Ganita (Mathematics) has been considered a very important subject since ancient times. We find very elaborate proof of this in Veda (which were compiled around 6000 BC). The concept of division, addition etc. was used even that time. Concepts of zero and infinite were there. We also find roots of algebra in *Veda*. When Indian Ganit reached Arab, they called it Algebra. Algebra was name of the Arabic book that described Indian concepts. This knowledge reached to Europe from there. And thus ancient Indian Ganit is currently referred to as Algebra.

The book Vedang jyotish (written 1000 BC) has mentioned the importance of Ganit as follows:-

Just as branches of a peacock and jewel-stone of a snake are placed at the highest place of body (forehead), similarly position of Ganit is highest in all the branches of Vedah and Shastras

Famous Jain Mathematician Mahaviracharya has said the following:-

What is the use of much speaking. Whatever object exists in this moving and nonmoving world, can not be understood without the base of Ganit(Mathematics).

This fact was well known to intellectuals of India that is why they gave special importance to the development of Mathematics, right from the beginning. When this knowledge was negligible in Arab and Europe, India had acquired great achievements.

People from Arab and other countries used to travel to India for commerce. While doing commerce, side by side, they also learnt easy to use calculation methods of India. Through them this knowledge reached to Europe. From time to time many inquisitive foreigners visited India and they delivered this matchless knowledge to their countries. This will not be exaggeration to say that till 12th century India was the World Guru in the area of Mathematics.

# Algebra — The Other Mathematics

In India around the 5^{th} century A.D. a system of mathematics that made astronomical calculations easy was developed. In those times its application was limited to astronomy as its pioneers were Astronomers. As tronomical calculations are complex and involve many variables that go into the derivation of unknown quantities. Algebra is a short-hand method of calculation and by this feature it scores over conventional arithmetic.

In ancient India conventional mathematics termed Ganitam was known before the development of algebra. This is borne out by the name - Bijaganitam, which was given to the algebraic form of computation. Bijaganitam means 'the other mathematics' (Bija means 'another' or 'second' and Ganitam means mathematics). The fact that this name was chosen for this system of computation implies that it was recognised as a parallel system of computation, different from the conventional one which was used since the past and was till then the only one. Some have interpreted the term Bija to mean seed, symbolizing origin or beginning. And the inference that Bijaganitam was the original form of computation is derived. Credence is lent to this view by the existence of mathematics in the Vedic literature which was also shorthand method of computation. But whatever the origin of algebra, it is certain that this technique of computation Originated in India and was current around 1500 years back. Aryabhatta an Indian mathematican who lived in the 5th century A.D. has referred to Bijaganitam in his treatise on Mathematics, Aryabhattiya. An Indian mathematician - astronomer, Bhaskaracharya has also authored a treatise on this subject. the treatise which is dated around the 12th century A.D. is entitled 'Siddhanta-Shiromani' of which one section is entitled Bijaganitam.

Thus the technique of algebraic computation was known and was developed in India in earlier times. From the 13th century onwards, India was subject to invasions from the Arabs and other Islamised communities like the Turks and Afghans. Alongwith these invader: came chroniclers and critics like Al-beruni who studied Indian society and polity.

The Indian system of mathematics could no have escaped their attention. It was also the age of the Islamic Renaissance and the Arabs generally improved upon the arts and sciences that they imbibed from the land they overran during their great Jehad. Th system of mathematics they observed in India was adapted by them and given the name 'Al-Jabr' meaning 'the reunion of broken parts'. 'Al' means 'The' & 'Jabr' mean 'reunion'. This name given by the Arabs indicates that they took it from an external source and amalgamated it with their concepts about mathematics.

Between the 10th to 13th centuries, the Christian kingdoms of Europe made numerous attempts to reconquer the birthplace of Jesus Christ from its Mohammedan-Arab rulers. These attempts called the Crusades failed in their military objective, but the contacts they created between oriental and occidental nations resulted in a massive exchange of ideas. The technique of algebr could have passed on to the west at thi time.

During the Renaissance in Europe, followed by the industrial revolution, the knowledge received from the east was further developed. Algebra as we know it today has lost any characteristics that betray it eastern origin save the fact that the tern 'algebra' is a corruption of the term 'Al jabr' which the Arabs gave to Bijaganitam Incidentally the term Bijaganit is still use in India to refer to this subject.

In the year 1816, an Englishman by the name James Taylor translated Bhaskara's Leelavati into English. A second English translation appeared in the following year (1817) by the English astronomer Henry Thomas Colebruke. Thus the works of this Indian mathematician astronomer were made known to the western world nearly 700 years after he had penned them, although his ideas had already reached the west through the Arabs many centuries earlier.

In the words of the Australian Indologist A.L. Basham (A.L. Basham; The Wonder That was India.) "… the world owes most to India in the realm of mathematics, which was developed in the Gupta period to a stage more advanced than that reached by any other nation of antiquity. The success of Indian mathematics was mainly due to the fact that Indians had a clear conception of the abstract number as distinct from the numerical quantity of objects or spatial extension."

Thus Indians could take their mathematical concepts to an abstract plane and with the aid of a simple numerical notation devise a rudimentary algebra as against the Greeks or the ancient Egyptians who due to their concern with the immediate measurement of physical objects remained confined to Mensuration and Geometry.

# Geometry and Algorithm

But even in the area of Geometry, Indian mathematicians had their contribution. There was an area of mathematical applications called Rekha Ganita (Line Computation). The Sulva Sutras, which literally mean 'Rule of the Chord' give geometrical methods of constructing altars and temples. The temples layouts were called Mandalas. Some of important works in this field are by Apastamba, Baudhayana, Hiranyakesin, Manava, Varaha and Vadhula.

The Arab scholar Mohammed Ibn Jubair al Battani studied Indian use of ratios from Retha Ganita and introduced them among the Arab scholars like Al Khwarazmi, Washiya and Abe Mashar who incorporated the newly acquired knowledge of algebra and other branches of Indian mathema into the Arab ideas about the subject.

The chief exponent of this Indo-Arab amalgam in mathematics was Al Khwarazmi who evolved a technique of calculation from Indian sources. This technique which was named by westerners after Al Khwarazmi as "Algorismi" gave us the modern term Algorithm, which is used in computer software.

Algorithm which is a process of calculation based on decimal notation numbers. This method was deduced by Khwarazmi from the Indian techniques geometric computation which he had st ied. Al Khwarazmi's work was translated into Latin under the title "De Numero Indico" which means 'of Indian Numerals' thus betraying its Indian origin. This translation which belong to the 12th century A.D credited to one Adelard who lived in a town called Bath in Britian.

Thus Al Khwarazmi and Adelard could looked upon as pioneers who transmit Indian numerals to the west. Incidents according to the Oxford Dictionary, word algorithm which we use in the English language is a corruption of the name Khwarazmi which literally means '(a person) from Khawarizm', which was the name of the town where Al Khwarazmi lived. To day unfortunately', the original Indian texts that Al Khwarazmi studied arelost to us, only the translations are avail able .

The Arabs borrowed so much from India the field of mathematics that even the subject of mathematics in Arabic came to known as Hindsa which means 'from India and a mathematician or engineer in Arabic is called Muhandis which means 'an expert in Mathematics'. The word Muhandis possibly derived from the Arabic term mathematics viz. Hindsa.

# The Concept of Zero

The concept of zero also originated inancient India. This concept may seem to be a very ordinary one and a claim to its discovery may be viewed as queer. But if one gives a hard thought to this concept it would be seen that zero is not just a numeral. Apart from being a numeral, it is also a concept, and a fundamental one at that. It is fundamental because, terms to identify visible or perceptible objects do not require much ingenuity.

But a concept and symbol that connotes nullity represents a qualitative advancement of the human capacity of abstraction. In absence of a concept of zero there could have been only positive numerals in computation, the inclusion of zero in mathematics opened up a new dimension of negative numerals and gave a cut off point and a standard in the measurability of qualities whose extremes are as yet unknown to human beings, such as temperature.

In ancient India this numeral was used in computation, it was indicated by a dot and was termed Pujyam. Even today we use this term for zero along with the more current term Shunyam meaning a blank. But queerly the term Pujyam also means holy. Param-Pujya is a prefix used in written communication with elders. In this case it means respected or esteemed. The reason why the term Pujya - meaning blank - came to be sanctified can only be guessed.

Indian philosophy has glorified concepts like the material world being an illusion Maya), the act of renouncing the material world (Tyaga) and the goal of merging into the void of eternity (Nirvana). Herein could lie the reason how the mathematical concept of zero got a philosophical connotation of reverence.

It is possible that like the technique of algebra; the concept of zero also reached the west through the Arabs. In ancient India the terms used to describe zero included Pujyam, Shunyam, Bindu the concept of a void or blank was termed as Shukla and Shubra. The Arabs refer to the zero as Siphra or Sifr from which we have the English terms Cipher or Cypher. In English the term Cipher connotes zero or any Arabic numeral. Thus it is evident that the term Cipher is derived from the Arabic Sifr which in turn is quite close to the Sanskrit term Shubra.

The ancient India astronomer Brahmagupta is credited with having put forth the concept of zero for the first time: Brahmagupta is said to have been born the year 598 A.D. at Bhillamala (today's Bhinmal ) in Gujarat, Western India. ] much is known about Brahmagupta's early life. We are told that his name as a mathematician was well established when K Vyaghramukha of the Chapa dyansty m him the court astronomer. Of his two treatises, Brahma-sputa siddhanta and Karanakhandakhadyaka, first is more famous. It was a corrected version of the old Astronomical text, Brahma siddhanta. It was in his Brahma-sphu siddhanta, for the first time ever had be formulated the rules of the operation zero, foreshadowing the decimal system numeration. With the integration of zero into the numerals it became possible to note higher numerals with limited charecters.

In the earlier Roman and Babylonian systems of numeration, a large number of chara acters were required to denote higher numerals. Thus enumeration and computation became unwieldy. For instance, as E the Roman system of numeration, the number thirty would have to be written as X: while as per the decimal system it would 30, further the number thirty three would be XXXIII as per the Roman system, would be 33 as per the decimal system. Thus it is clear how the introduction of the decimal system made possible the writing of numerals having a high value with limited characters. This also made computation easier.

Apart from developing the decimal system based on the incorporation of zero in enumeration, Brahmagupta also arrived at solutions for indeterminate equations of 1 type ax2+1=y2 and thus can be called the founder of higher branch of mathematics called numerical analysis. Brahmagupta's treatise Brahma-sputa-siddhanta was translated into Arabic under the title Sind Hind).

For several centuries this translation mained a standard text of reference in the Arab world. It was from this translation of an Indian text on Mathematics that the Arab mathematicians perfected the decimal system and gave the world its current system of enumeration which we call the Arab numerals, which are originally Indian numerals.

# Beginning on Indian Mathematics

The auspicious beginning on Indian Mathematics is in Aadi Granth (ancient/eternal book) Rigved. The history of Indian Mathematics can be divided into 5 parts, as following.

1) Ancient Time (Before 500 BC)

- Vedic Time (1000 BC-At least 6000 BC)
- Later Vedic Time (1000 BC-500BC)

2) Pre Middle Time (500 BC- 400 AD)

3) Middle Time or Golden Age (400 AD - 1200 AD)

4) Later Middle Time (1200 AD - 1800 AD)

5) Current Time (After 1800 AD)

## 1. Ancient Time (Before 500 BC)

Ancient time is very important in the history of Indian Mathematics. In this time different branches of Mathematics, such as Numerical Mathematics; Algebra; Geometrical Mathematics, were properly and strongly established.

There are two main divisions in Ancient Time. Numerical Mathematics developed in Vedic Time and Geometrical Mathematics developed in Later Vedic Time.

1a) Vedic Time (1000 BC-At least 6000 BC)

Numerals and decimals are cleanly mentioned in Vedah (Compiled at lease 6000 BC). There is a Richa in Veda, which says the following-

In the above Richa , Dwadash (12), Treeni (2), Trishat (300) numerals have been used. This indicates the use of writing numerals based on 10.

In this age the discovery of ZERO and "10th place value method"(writing number based on 10) is great contribution to world by India in the arena of Mathematics.

If "zero" and "10 based numbers" were not discovered, it would not have been possible today to write big numbers.

The great scholar of America Dr. G. B. Halsteed has also praised this. Shlegal has also accepted that this is the second greatest achievement of human race after the discovery of Alphabets.

This is not known for certain that who invented "zero" and when. But it has been in use right from the "vedic" time. The importance of "zero" and "10th place value method" is manifested by their wide spread use in today's world. This discovery is the one that has helped science to reach its current status.

In the second section of earlier portion of Narad Vishnu Puran (written by Ved Vyas) describes "mathematics" in the context of Triskandh Jyotish. In that numbers have been described which are ten times of each other, in a sequence (10 to the power n). Not only that in this book, different methods of "mathematics" like Addition, Subtraction, Multiplication, Addition, Fraction, Square, Square root, Cube root et-cetera have been elaborately discussed. Problems based on these have also been solved.

This proves at that time various mathematical methods were not in concept stage, rather those were getting used in a methodical and expanded manner.

"10th place value method" dispersed from India to Arab. From there it got transferred to Western countries. This is the reason that digits from 1-9 are called "hindsa" by the people of Arab. In western countries 0,1,2,3,4,5,6,7,8,9 are called Hindu-Arabic Numerals.

### 1b. Later Vedic Time (1000 BC - 500 BC)

#### 1b.1. Shulv and Vedang Jyotish Time

Vedi was very important while performing rituals. On the top of "Vedi" different type of geomit (geometry: as you notice this word is derived from a Sanskrit word) were made. To measure those geometry properly, "geometrical mathematics" was developed. That knowledge was available in form of Shulv Sutras (Shulv Formulae). Shulv means rope. This rope was used in measuring geometry while making vedis.

In that time we had three great formulators-Baudhayan, Aapstamb and Pratyayan. Apart from them Manav, Matrayan, Varah and Bandhul are also famous mathematician of that time.

#### 1b.2. Surya Pragyapti Time

We find elaborated description of Mathematics in the Jain literature. In fact the clarity and elaboration by which Mathematics is described in Jain literature, indicates the tendency of Jain philosophy to convey the knowledge to the language and level of common people (This is in deviation to the style of Veda which told the facts indirectly).

Surya Pragyapti and Chandra Pragyapti (At least 500 BC) are two famous scriptures of Jain branch of Ancient India. These describe the use of Mathematics.

Deergha Vritt (ellipse) is clearly described in the book titled Surya Pragyapti. "Deergha Vritt" means the outer circle (Vritta) on a rectangle(Deergha), that was also known as Parimandal. This is clear that Indians had discovered this at least 150 years before Minmax (150 BC). As this history was not known to the West so they consider Minmax as the first time founder of ellipse.

This is worth mentioning that in the book Bhagvati Sutra (Before 300 BC) the word Parimandal has been used for Deergha Vritt (ellipse). It has been described to have two types 1) Pratarparimandal and 2)Ghanpratarparimandal.

Jain Aacharyas contributed a lot in the development of Mathematics. These gurus have described different branches of mathematics in a very through and interesting manner. They are examples too.

They have described fractions, algebraic equations, series, set theory, logarithm, and exponents …. Under the set theory they have described with examples- finite, infinite, single sets. For logarithm they have used terms like Ardh Aached , Trik Aached, Chatur Aached. These terms mean log base 2, log base 3 and log base 4 respectively. Well before Joan Napier (1550-1617 AD), logarithm had been invented and used in India which is a universal truth.

Buddha literature has also given due importance to Mathematics. They have divided Mathematics under two categories- 1) Garna (Simple Mathematics) and 2)Sankhyan (Higher Mathematics). They have described numbers under three categories-1)Sankheya(countable),2)Asankheya(uncountable) and 3)Anant(infinite). Which clearly indicates that Indian Intellectuals knew "infinite number" very well.

## 2. Pre Middle Time (500 BC- 400 AD)

This is unfortunate that except for the few pages of the books Vaychali Ganit, Surya Siddhanta and Ganita Anoyog of this time, rest of the writings of this time are lost. From the remainder pages of this time and the literature of Aryabhatt, Brahamgupt et-cetera of Middle Time, we can conclude that in this time too Mathematics underwent sufficient development.

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

The book titled Vaychali Ganit discusses in detail the following -the basic calculations of mathematics, the numbers based on 10, fraction, square, cube, rule of false position, interest methods, questions on purchase and sale… The book has given the answers of the problems and also described testing methods. Vachali Ganit is a proof of the fact that even at that time (300 BC) India was using various methods of the current Numerical Mathematics. This is noticeable that this book is the only written Hindu Ganit book of this time that was found as a few survived pages in village Vaychat Gram (Peshawar) in 1000 AD.

Sathanang Sutra has mentioned five types of infinite and Anoyogdwar Sutra has mentioned four types of Pramaan (Measure). This Granth(book) has also described permutations and combinations which are termed as Bhang and Vikalp .

This is worth mentioning that in the book Bhagvati Sutra describes the following. From n types taking 1-1,2-2 types together the combinations such made are termed as Akak, Dwik Sanyog and the value of such combinations is mentioned as n(n-1)/2 which is used even today.

Roots of the Modern Trignometry lie in the book titled Surya Siddhanta . It mentions Zya(Sine), Otkram Zya(Versesine), and Kotizya(Cosine). Please remember that the same word (Zia) changed to "Jaib" in Arab. The translation of Jaib in Latin was done as "Sinus". And this "Sinus" became "Sine" later on.

This is worth mentioning that Trikonmiti word is pure Indian and with the time it changed to Trignometry. Indians used Trignometry in deciding the position , motion et-cetera of the spatial planets.

In this time the expansion of Beezganit (When this knowledge reached Arab from India it became Algebra)was revolutionary. The roots of Modern Algebra lie in the book Vaychali Ganit. In this book while describing Isht KarmaIsht Karm "Rule of False" as the origin of expansion of Algebra. Thus Algebra is also gifted to world by Indians

Although almost all ancient countries used quantities of unknown values and using them found the result of Numerical Mathematics. However the the expansion of Beez Ganit (Now known as Alzebra) became possible when right denotion method was developed. The glory for this goes to Indians who for the first time used Sanskrit Alphabet to denote unknown quantities. Infact expansion of Beez Ganit (Now known as Alzebra) became possible when Indians realized that all the calculations of Numerical Mathematics could be done by notations. And that +, - these signs can be used with those notations.

Indians developed rules of addition, subtraction, multiplication with these signs (+,-,x). In this context we can not forget the contribution of great mathematician Brahmgupt (628 AD). He said-

The multiplication of a positive number with a negative number comes out to be a negative number and multiplication of a positive number with a positive number comes out to be a positive number.

He further told:

When a positive number is divided by a positive number the result is a positive number and when a positive number is divided by a negative number or a negative number is divided by a positive number the result is a negative number.

Indians used notations for squares, cube and other exponents of numbers. Those notations are used even today in the mathematics. They gave shape to Beezganit Samikaran(Algebraic Equations). They made rules for transferring the quantities from left to right or right to left in an equation. Right from the 5th century AD, Indians majorly used aforementioned rules.

In the book titled Anoyogdwar Sutra has described some rules of exponents in Beez Ganit (Later the name Algebra became more popular).

It is without doubt that like Aank Ganit (Numerical Mathematics) Beez Ganit (Later the name Algebra became more popular) reached Arab from India. Arab mathematician Al-Khowarizmi (780-850 AD) has described topics based on Indian Beez Ganit in his book titled "Algebr". And when it reached Europe it was called Algebra.

As for as other countries are concerned we find that in the golden time of Greece Mathematics there was no sign of Algebra with respect to modern concept of Algebra. In classical period Greece people had ability to solve tough questions of Beez Ganit (Later the name Algebra became more popular) but there all solutions were based on Geometrical Mathematics. For the first time in Greece world, the concept of Beez Ganit (Later the name Algebra became more popular) is described in a books of Diofantus (275 AD). By that time Indians were far ahead. This is worth noting that the shape and form of current Beez Ganit (Later the name Algebra became more popular) is originally Indian.

## 3. Middle Time or Golden Age 400 AD- 1200 AD)

This period is called golden age of Indian Mathematics. In this time great mathematicians like Aryabhatt, Brahmgupt, Mahaveeracharya, Bhaskaracharya who gave a broad and clear shape to almost all the branches of mathematics which we are using today. The principles and methods which are in form of Sutra(formulae) in Vedas were brought forward with their full potential, in front of the common masses. To respect this time India gave the name "Aryabhatt" to its first space satellite.

The following is the description about great mathematicians and their creations.

## Aryabhatt (First) (490 AD)

He was a resident of Patna in India. He has described, in a very crisp and concise manner, the important fundamental principles of Mathematics only in 332 Shlokas. His book is titled Aryabhattiya. In the first two sections of Aryabhattiya, Mathematics is described. In the last two sections of Aryabhattiya, Jyotish (Astrology) is described. In the first section of the book, he has described the method of denoting big decimal numbers by the alphabets.

In the second section of the book Aryabhattiya we find difficult questions from topics such as Numerical Mathematics, Geometrical Mathematics, Trignometry and Beezganit (Algebra). He also worked on indeterminate equations of Beezganit (Later in West it was called Algebra). He was the first to use Vyutkram Zia (Which was later known as Versesine in the West) in Trignometry. He calculated the value of pi correct upto four decimal places.

He was first to find that the sun is stationary and the earth revolves around it. 1100 years later, this fact was accepted by Coppernix of West in 16th century. Galileo was hanged for accepting this.

### Bhaskar (First) (600 AD)

He did matchless work on Indeterminate equations. He expanded the work of Aryabhatt in his books titled Mahabhaskariya, Aryabhattiya Bhashya and Laghu Bhaskariya .

### Brahmgupt (628 AD)

His famous work is his book titled Brahm-sfut. This book has 25 chapters. In two chapters of the book, he has elaborately described the mathematical principles and methods. He threw light on around 20 processes and behavior of Mathematics. He described the rules of the solving equations of Beezganit (Algebra). He also told the solution of indeterminate equations with two exponent. Later Ailer in 1764 AD and Langrez in 1768 described the same.

Brahmgupt told the method of calculating the volume of Prism and Cone. He also described how to sum a GP Series. He was the first to tell that when we divide any positive or negative number by zero it becomes infinite.

### Mahaveeracharya (850 AD)

He wrote the book titled "Ganit Saar Sangraha". This book is on Numerical Mathematics. He has described the currently used method of calculating Least Common Multiple (LCM) of given numbers. The same method was used in Europe later in 1500 AD. He derived formulae to calculate the area of ellipse and quadrilateral inside a circle.

### Shridharacharya (850 AD)

He wrote books titled "Nav Shatika", "Tri Shatika", "Pati Ganit". These books are on Numerical Mathematics. His books on Beez Ganit (Algebra) are lost now, but his method of solving quadratic equations is still used. This is method is also called "Shridharacharya Niyam". The great thing is that currently we use the same formula as told by him. His book titled "Pati Ganit" has been translated into Arabic by the name "Hisabul Tarapt".

### Aryabhatta Second (950 AD)

He wrote a book titled Maha Siddhanta. This book discusses Numerical Mathematics (Ank Ganit) and Algebra. It describes the method of solving algebraic indeterminate equations of first order. He was the first to calculate the surface area of a sphere. He used the value of pi as 22/7.

### Shripati Mishra (1039 AD)

He wrote the books titled Siddhanta Shekhar and Ganit Tilak. He worked mainly on permutations and combinations. Only first section of his book Ganit Tilak is available.

### Nemichandra Siddhanta Chakravati (1100 AD)

His famous book is titled Gome-mat Saar. It has two sections. The first section is Karma Kaand and the second section is titled Jeev Kaand. He worked on Set Theory. He described universal sets, all types of mapping, Well Ordering Theorems et-cetera.One to One Mapping was used by Gailileo and George Kanter(1845-1918) after many centuries.

### Bhaskaracharya Second (1114 AD)

He has written excellent books namely Siddhanta Shiromani,Leelavati Beezganitam,Gola Addhaya,Griha Ganitam and Karan Kautoohal. He gave final touch to Numerical Mathematics, Beez Ganit (Algebra), and Trikonmiti (Trignometry).

The concepts which were in the form of formulae in Vedah. He has also described 20 methods and 8 behaviors of Brahamgupt.

Great Hankal has praised a lot Bhaskaracharya's Chakrawaat Method of solving indeterminate equations of Beezganit (Algebra). This Bhaskaracharya's Chakrawaat Method was used by Ferment in 1667 to solve indeterminate equations.

In his book Siddhanta Shiromani, he has described in length the concepts of Trignometry. He has described Sine, Cosine, Versesine,… Infinitesimal Calculus and Integration. He wrote that earth has gravitational force.

## 4. Later Middle Period (1200 AD- 1800 AD)

Not much original work was done after Bhaskaracharya Second. Comments on ancient texts are the main contribution of this period.

### Narayan Pundit (1356 AD)

He wrote the book titled Ganit Kaumidi. This book deals with Permutations and Combinations, Partition of Numbers, Magic Squares.

### Neel Kanta (1587 AD)

He wrote the book titled Tagikani Kanti. This book deals with Zeotish Ganit(Astrological Mathematics).

### Kamalakar (1608 AD)

He wrote a book titled Siddhanta Tatwa Viveka.

### Samraat Jagannath (1731 AD)

He wrote two books titled Samraat Siddhanta and Rekha Ganit (Line Mathematics)

Apart from the above-mentioned mathematicians we have a few more worth mentioning mathematicians. From Kerla we have Madhav (1350-1410 AD). Jyeshta Deva (1500-1610 AD) wrote a book titled Ukti Bhasha. Shankar Paarshav (1500-1560 AD) wrote a book titled Kriya Kramkari.

## 5. Current Period (1800 AD- Current)

Please find below a list of famous mathematicians and their writings.

### Nrisingh Bapudev Shastri (1831 AD)

He wrote books on Geometrical Mathematics, Numerical Mathematics and Trignometry.

### Sudhakar Dwivedi (1831 AD)

He wrote books titled Deergha Vritta Lakshan(which means characteristics of ellipse), Goleeya Rekha Ganit(which means sphere line mathematics),Samikaran Meemansa(which means analysis of equations) and Chalan Kalan.

## Ramanujam (1889 AD)

Ramanujam is a modern mathematics scholar. He followed the vedic style of writing mathematical concepts in terms of formulae and then proving it. His intellectuality is proved by the fact it took all mettle of current mathematicians to prove a few out of his total 50 theorems.

### Swami Bharti Krishnateerthaji Maharaj (1884-1960 AD)

He wrote the book titled Vedic Ganit.

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