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A Brief History of Indian Mathematics
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Prof. Vijaya Kumar Murty  
Prof. Vijaya Kumar Murty (September 2007 (517)
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Mathematics has played a signifcant role in the development of Indian culture for millennia. Mathematical ideas that originated in the Indian subcontinent have had a profound impact on the world. Swami Vivekananda said: ‘You know how many sciences had their origin in India. Mathematics began there. You are even today counting 1, 2, 3, etc. to zero, afer
+
Mathematics has played a significant role in the development of Indian culture for millennia. Mathematical ideas that originated in the Indian subcontinent have had a profound impact on the world. Swami Vivekananda said: ‘You know how many sciences had their origin in India. Mathematics began there. You are even today counting 1, 2, 3, etc. to zero, after Sanskrit figures, and you all know that algebra also originated in India.’1  
Sanskrit fgures, and you all know that algebra also originated in India.’1  
+
It  is also a ftting time to review the contribu-tions of Indian mathematicians from ancient times to the present, as in 2010, India will be hosting the International Congress of Mathematicians. Tis quadrennial meeting brings together mathemati-cians from around the world to discuss the most signifcant developments  in the subject over the past four years and to get a sense of where the sub-ject is heading in the next four. Te idea of holding such a congress at regular intervals actually started at the Columbian Exhibition in Chicago in 1893.
+
Tis exhibition had sessions to highlight the ad-vancement of knowledge in diferent felds. One
+
of these was a session on mathematics. Another, perhaps more  familiar  to readers of Prabuddha Bharata, was the famous Parliament of Religions in which Swami Vivekananda frst made his public appearance in the West.
+
Following the Chicago meeting, the frst Inter-national Congress of Mathematicians took place in Zurich in 1897. It was at the next meeting at Paris in 1900 that Hilbert formulated his now famous 23 problems. Since that time, the congress has been meeting approximately every four years  in difer-ent cities around the world, and in 2010, the venue will be Hyderabad, India. Tis is the frst time in its more than hundred-year history that the con-gress will be held in India. Tis
+
meeting could serve as an impe-tus and stimulus to mathematical thought in the subcontinent, pro-vided the community is prepared for it. Preparation would  largely consist in being aware of the tra-dition of mathematics  in  India, from ancient times to modern, and in embracing
+
the potential and possibility of developing this tra-dition to new heights in the coming millennia.
+
In ancient times, mathematics was mainly used in an auxiliary or applied role. Tus, mathemati-cal methods were used to solve problems in archi-tecture and construction (as in the public works of the Harappan civilization),  in astronomy and astrology (as in the works of the Jain mathemati-cians), and in the construction of Vedic altars (as in the case of the Shulba Sutras of Baudhayana and his successors). By the sixth or ffh century BCE, mathematics was being studied for  its own sake. In modern times, mathematics is studied both for its own sake, as well as for its applications in other felds of knowledge.
+
Te aim of this article is to give a brief review of a few of the outstanding innovations introduced by Indian mathematics from ancient times to modern. As we shall see, there does not seem to have been a time in Indian history when mathematics was not being developed. Recent work has unearthed many manuscripts, and what were previously regarded as dormant periods in Indian mathematics are now known to have been very active. Even a small study
+
of this subject leaves one with a sense of wonder at the depth and breadth of ancient Indian thought.
+
The picture  is not yet complete, and  it seems that there is much work to do in the feld of the
+
  
  
history of Indian mathematics. Te challenges are two-fold. First, there is the task of locating and identifying manuscripts and of translating them into a language that is more familiar to modern scholars. Second, there is the task of interpreting the signifcance of the work that was done.  
+
It is also a fitting time to review the contributions of Indian mathematicians from ancient times to the present, as in 2010, India will be hosting the International Congress of Mathematicians. This quadrennial meeting brings together mathematicians from around the world to discuss the most significant developments in the subject over the past four years and to get a sense of where the subject is heading in the next four. The idea of holding such a congress at regular intervals actually started at the Columbian Exhibition in Chicago in 1893. This exhibition had sessions to highlight the advancement of knowledge in different fields. One of these was a session on mathematics. Another, perhaps more familiar to readers of Prabuddha Bharata, was the famous Parliament of Religions in which Swami Vivekananda first made his public appearance in the West.
Since much of the past work in this area has tended to adopt a Eurocentric perspective and in-
+
 
terpretation, it is necessary to take a fresh, objec-tive look. Te time is ripe to make a major efort to develop as complete a picture as possible of Indian mathematics. Tose who are interested in embark-ing on such an efort can fnd much helpful mate-rial online.2
+
 
We may ask what the term ‘Indian’ means in the context of this discussion. Mostly, it refers to the In-dian subcontinent, but for more recent history we include also the diaspora and people whose roots can be traced to the Indian subcontinent, wherever they may be geographically located.  
+
Following the Chicago meeting, the first Inter-national Congress of Mathematicians took place in Zurich in 1897. It was at the next meeting at Paris in 1900 that Hilbert formulated his now famous 23 problems. Since that time, the congress has been meeting approximately every four years in different cities around the world, and in 2010, the venue will be Hyderabad, India. This is the first time in its more than hundred-year history that the congress will be held in India. This meeting could serve as an impetus and stimulus to mathematical thought in the subcontinent, pro-vided the community is prepared for it. Preparation would largely consist in being aware of the tradition of mathematics in India, from ancient times to modern, and in embracing the potential and possibility of developing this tradition to new heights in the coming millennia.
Mathematics in Ancient Times
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(3000 to 600 BCE)
+
 
Te Indus valley civilization is considered to have ex-isted around 3000 BCE. Two of its most famous cit-ies, Harappa and Mahenjo-daro, provide evidence that construction of buildings followed a stand-ardized measurement which was decimal in nature. Here, we see mathematical ideas developed for the purpose of construction. This civilization had an ad-
+
In ancient times, mathematics was mainly used in an auxiliary or applied role. Thus, mathematical methods were used to solve problems in architecture and construction (as in the public works of the Harappa civilization), in astronomy and astrology (as in the works of the Jain mathematicians), and in the construction of Vedic altars (as in the case of the Shula Sutras of Baudhayana and his successors). By the sixth or fifth century BCE, mathematics was being studied for its own sake. In modern times, mathematics is studied both for its own sake, as well as for its applications in other fields of knowledge.
vanced brick-making technology (having invented the kiln). Bricks were used in the construction of buildings and embankments for food control.  
+
 
The study of astronomy is considered to be even older, and there must have been mathematical the-ories on which it was based. Even in later times, we fnd that astronomy motivated considerable mathematical development, especially in the feld of trigonometry.
+
 
Much has been written about the mathemati-cal constructions that are to be found in Vedic lit-
+
The aim of this article is to give a brief review of a few of the outstanding innovations introduced by Indian mathematics from ancient times to modern. As we shall see, there does not seem to have been a time in Indian history when mathematics was not being developed. Recent work has unearthed many manuscripts, and what were previously regarded as dormant periods in Indian mathematics are now known to have been very active. Even a small study of this subject leaves one with a sense of wonder at the depth and breadth of ancient Indian thought.
erature. In particular, the Shatapatha Brahmana, which is a part of the Shukla Yajur Veda, contains detailed descriptions of the geometric construc-tion of altars for yajnas. Here, the brick-making technology of the Indus valley civilization was put to a new use. As usual, there are diferent interpre-tations of the dates of Vedic texts, and in the case of this Brahmana, the range is from 1800 to about 800 BCE. Perhaps it is even older. Supplementary to the Vedas are the Shulba Sutras. Tese texts are considered to date from 800 to 200 BCE. Four in number, they are named afer their authors: Baudhayana (800 BCE), Manava (750 BCE), Apastamba (600 BCE), and Katyayana (200 BCE). Te sutras contain the famous theorem commonly attributed to Pythagoras. Some scholars (such as Seidenberg) feel that this theorem is al-
+
 
ready present in the Shatapatha Brahmana. Much later, Bhaskaracharya gave an algebraic proof of this theorem, as opposed to the geometric proof that the Greeks, and possibly the Chinese, were aware of.3  
+
 
The Shulba Sutras introduce the concept of ir-rational numbers, numbers that are not the ratio of two whole numbers. For example, the square root of 2 is one such number. Te sutras give a way of approximating the square root of a number using rational numbers through a recursive procedure which in modern language would be a ‘series ex-pansion’. Tis predates, by far, the European use of Taylor series.
+
The picture is not yet complete, and it seems that there is much work to do in the field of the history of Indian mathematics. The challenges are two-fold. First, there is the task of locating and identifying manuscripts and of translating them into a language that is more familiar to modern scholars. Second, there is the task of interpreting the significance of the work that was done.  
It is interesting that the mathematics of this period seems to have been developed for solving practical geometric problems, especially the con-struction of religious altars. However, the study of the series expansions for certain functions already hints at the development of an algebraic perspec-tive. In later times, we fnd a shif towards algebra, with simplifcation of algebraic formulae and sum-mation of series acting as catalysts for mathemati-cal discovery.
+
 
Jain Mathematics (600 BCE to 500 CE)
+
 
Tis is a topic that scholars have started studying only recently. Knowledge of this period of math-ematical history is still fragmentary, and it is a fer-tile area for future scholarly studies. Just as Vedic philosophy and theology stimulated the develop-ment of certain aspects of mathematics, so too did the rise of Jainism. Jain cosmology led to ideas of the infnite. Tis, in turn, led to the development of the notion of ‘orders of infnity’ as a mathematical concept. By ‘orders of infnity’, we mean a theory by which one set could be deemed to be ‘more in-fnite’ than another. In modern language, this cor-responds to the notion of cardinality. For a fnite
+
Since much of the past work in this area has tended to adopt a Eurocentric perspective and interpretation, it is necessary to take a fresh, objective look. The time is ripe to make a major effort to develop as complete a picture as possible of Indian mathematics. Those who are interested in embarking on such an effort can find much helpful material online.2
set, its cardinality is the number of elements it con-tains. However, we need a more sophisticated no-tion to measure the size of an infnite set. In Europe,it was not until Cantor’s work in the nineteenth century that a proper concept of cardinality was established.
+
 
Besides the investigations into infnity, this period saw developments in several other felds
+
 
such as number theory, geometry, computing with fractions, and combinatorics. In particular, the recursion formula for binomial coefcients and the ‘Pascal’s triangle’ were already known in this period.
+
We may ask what the term ‘Indian’ means in the context of this discussion. Mostly, it refers to the Indian subcontinent, but for more recent history we include also the diaspora and people whose roots can be traced to the Indian subcontinent, wherever they may be geographically located.  
As mentioned in the previous section, astrono-my had been studied in India since ancient times. Tis subject is ofen confused with astrology. Swa-mi Vivekananda has speculated that astrology came to India from the Greeks and that astronomy was borrowed by the Greeks from India. Indirect evi-dence for this is provided by a text by Yavaneshvara (c. 200 CE) which popularized a Greek astrology text dating back to 120 BCE.  
+
 
The period 600 BCE to 900 CE coincides with the rise and dominance of Buddhism. In the La-
+
 
litavistara, a biography of the Buddha which may have been written around the frst  century CE, there is an incident about Gautama being asked to state the name of large powers of 10, starting with 107. He is able to give names to numbers up to 1053 (tallakṣaṇa). Te very fact that such large numbers had names suggests that the mathematicians of the day were comfortable thinking about very large numbers. It is hard to imagine calculat-ing with such numbers without some form of place-value system.  
+
== Mathematics in Ancient Times (3000 to 600 BCE) ==
Brahmi Numerals,
+
 
the Place-value System, and Zero
+
 
No account of Indian mathematics would be com-plete without a discussion of Indian numerals, the place-value system, and the concept of zero. The numerals that we use even today can be traced to the Brahmi numerals that seem to have made their appearance in 300 BCE. But Brahmi numerals were not part of a place-value system. Tey evolved into  
+
The Indus valley civilization is considered to have existed around 3000 BCE. Two of its most famous cities, Harappa and Mohenjo-Daro, provide evidence that construction of buildings followed a standardized measurement which was decimal in nature. Here, we see mathematical ideas developed for the purpose of construction. This civilization had an advanced brick-making technology (having invented the kiln). Bricks were used in the construction of buildings and embankments for food control.  
the Gupta numerals around 400 CE and subse-quently into the Devanagari numerals, which de-veloped slowly between 600 and 1000 CE.  
+
 
By 600 CE, a place-value decimal system was well in use in India. Tis means that when a number is written down, each symbol that is used has an absolute value, but also a value relative to its posi-tion. For example, the numbers 1 and 5 have a value on their own, but also have a value relative to their position in the number 15. Te  importance of a place-value system need hardly be emphasized. It would sufce to cite an of-quoted remark by La-place: ‘It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ig-nore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the frst rank of useful inventions; and we shall appreciate the grandeur of the achieve-ment the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.’4
+
 
A place-value system of numerals was apparently known in other cultures; for example, the Babylo-nians used a sexagesimal place-value system as early as 1700 BCE, but the Indian system was the frst decimal system. Moreover, until 400 BCE, the Ba-bylonian system had an inherent ambiguity as there was no symbol for zero. Tus, it was not a complete place-value system in the way we think of it today.  
+
The study of astronomy is considered to be even older, and there must have been mathematical theories on which it was based. Even in later times, we find that astronomy motivated considerable mathematical development, especially in the field of trigonometry.
The elevation of zero to the same status as other numbers involved difculties that many brilliant mathematicians struggled with. Te main problem as that the rules of arithmetic had to be formu-lated so as to include zero. While addition, subtraction, and multiplication with zero were mastered, division was a more subtle question. Today, we know that division by zero is not well-defned and so has to be excluded from the rules of arithmetic. But this understanding did not come all at once, and took the combined eforts of many minds. It is interesting to note that it was not until the seven-teenth century that zero was being used in Europe, and the path of mathematics from India to Europe is the subject of much historical research.  
+
 
The Classical Era of Indian Mathematics  
+
 
(500 to 1200 CE)
+
Much has been written about the mathematical constructions that are to be found in Vedic literature. In particular, the Shatapatha Brahmana, which is a part of the Shukla Yajur Veda, contains detailed descriptions of the geometric construction of altars for yajnas. Here, the brick-making technology of the Indus valley civilization was put to a new use. As usual, there are different interpretations of the dates of Vedic texts, and in the case of this Brahman, the range is from 1800 to about 800 BCE. Perhaps it is even older.  
Te most famous names of Indian mathematics belong to what is known as the classical era. Tis
+
 
includes Aryabhata I (500 CE), Brahmagupta (700 CE), Bhaskara I (900 CE), Mahavira (900 CE), Aryabhata II (1000 CE), and Bhaskaracharya or Bhaskara II (1200 CE).
+
 
During this period, two centers of mathemati-cal research emerged, one at Kusumapura near Pa-taliputra and the other at Ujjain. Aryabhata I was the dominant fgure at Kusumapura and may even have been the founder of the local school. His fun-damental work, the Aryabhatiya, set the agenda for research in mathematics and astronomy in India for many centuries.  
+
Supplementary to the Vedas are the Shula Sutras. These texts are considered to date from 800 to 200 BCE. Four in number, they are named after their authors: Baudhayana (800 BCE), Manava (750 BCE), Apastamba (600 BCE), and Katyayana (200 BCE). The sutras contain the famous theorem commonly attributed to Pythagoras. Some scholars (such as Seidenberg) feel that this theorem is already present in the Shatapatha Brahmana. Much later, Bhaskaracharya gave an algebraic proof of this theorem, as opposed to the geometric proof that the Greeks, and possibly the Chinese, were aware of.3  
One of Aryabhata’s discoveries was a method for solving linear equations of the form ax + by = c.Here a, b, and c are whole numbers, and we are seek-ing values of x and y in whole numbers satisfying the above equation. For example, if a = 5, b = 2, and c = 8, then x = 8 and y = −16 is a solution. In fact, there are infnitely many solutions:
+
 
 +
 
 +
The Shulba Sutras introduce the concept of irrational numbers, numbers that are not the ratio of two whole numbers. For example, the square root of 2 is one such number. The sutras give a way of approximating the square root of a number using rational numbers through a recursive procedure which in modern language would be a ‘series expansion’. This predates, by far, the European use of Taylor series.
 +
 
 +
 
 +
It is interesting that the mathematics of this period seems to have been developed for solving practical geometric problems, especially the construction of religious altars. However, the study of the series expansions for certain functions already hints at the development of an algebraic perspective. In later times, we find a shift towards algebra, with simplification of algebraic formulae and summation of series acting as catalysts for mathematical discovery.
 +
 
 +
 
 +
== Jain Mathematics (600 BCE to 500 CE) ==
 +
 
 +
 
 +
This is a topic that scholars have started studying only recently. Knowledge of this period of mathematical history is still fragmentary, and it is a fertile area for future scholarly studies. Just as Vedic philosophy and theology stimulated the development of certain aspects of mathematics, so too did the rise of Jainism. Jain cosmology led to ideas of the infinite. This, in turn, led to the development of the notion of ‘orders of infinity’ as a mathematical concept. By ‘orders of infinity’, we mean a theory by which one set could be deemed to be ‘more infinite’ than another. In modern language, this corresponds to the notion of cardinality. For a finite set, its cardinality is the number of elements it contains. However, we need a more sophisticated notion to measure the size of an infinite set. In Europe, it was not until Cantor’s work in the nineteenth century that a proper concept of cardinality was established.
 +
 
 +
 
 +
Besides the investigations into infinity, this period saw developments in several other fields such as number theory, geometry, computing with fractions, and combinatorics. In particular, the recursion formula for binomial coefficients and the ‘Pascal’s triangle’ were already known in this period  
 +
 
 +
 
 +
As mentioned in the previous section, astronomy had been studied in India since ancient times. This subject is often confused with astrology. Swami Vivekananda has speculated that astrology came to India from the Greeks and that astronomy was borrowed by the Greeks from India. Indirect evidence for this is provided by a text by Yavaneshvara (c. 200 CE) which popularized a Greek astrology text dating back to 120 BCE.  
 +
 
 +
 
 +
The period 600 BCE to 900 CE coincides with the rise and dominance of Buddhism. In the Lalitavistara, a biography of the Buddha which may have been written around the first century CE, there is an incident about Gautama being asked to state the name of large powers of 10, starting with 107. He is able to give names to numbers up to 1053 (tallaksana). The very fact that such large numbers had names suggests that the mathematicians of the day were comfortable thinking about very large numbers. It is hard to imagine calculating with such numbers without some form of place-value system.  
 +
 
 +
 
 +
== Brahmi Numerals, the Place-value System, and Zero ==
 +
 
 +
 
 +
No account of Indian mathematics would be complete without a discussion of Indian numerals, the place-value system, and the concept of zero. The numerals that we use even today can be traced to the Brahma numerals that seem to have made their appearance in 300 BCE. But Brahma numerals were not part of a place-value system. They evolved into the Gupta numerals around 400 CE and subsequently into the Devanagari numerals, which developed slowly between 600 and 1000 CE.  
 +
 
 +
 
 +
By 600 CE, a place-value decimal system was well in use in India. This means that when a number is written down, each symbol that is used has an absolute value, but also a value relative to its position. For example, the numbers 1 and 5 have a value on their own, but also have a value relative to their position in the number 15. The importance of a place-value system need hardly be emphasized. It would suffice to cite an oft-quoted remark by La-place: ‘It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.’4
 +
 
 +
 
 +
A place-value system of numerals was apparently known in other cultures; for example, the Babylonians used a sexagesimal place-value system as early as 1700 BCE, but the Indian system was the first decimal system. Moreover, until 400 BCE, the Ba-bylonian system had an inherent ambiguity as there was no symbol for zero. Thus, it was not a complete place-value system in the way we think of it today.  
 +
 
 +
 
 +
The elevation of zero to the same status as other numbers involved difficulties that many brilliant mathematicians struggled with. The main problem was that the rules of arithmetic had to be formulated so as to include zero. While addition, subtraction, and multiplication with zero were mastered, division was a more subtle question. Today, we know that division by zero is not well-defined and so has to be excluded from the rules of arithmetic. But this understanding did not come all at once, and took the combined eforts of many minds. It is interesting to note that it was not until the seventeenth century that zero was being used in Europe, and the path of mathematics from India to Europe is the subject of much historical research.  
 +
 
 +
 
 +
== The Classical Era of Indian Mathematics (500 to 1200 CE) ==
 +
 
 +
 
 +
The most famous names of Indian mathematics belong to what is known as the classical era. This includes Aryabhata I (500 CE), Brahmagupta (700 CE), Bhaskara I (900 CE), Mahavira (900 CE), Aryabhata II (1000 CE), and Bhaskaracharya or Bhaskara II (1200 CE).  
 +
 
 +
 
 +
During this period, two centers of mathematical research emerged, one at Kusumapura near Pataliputra and the other at Ujjain. Aryabhata I was the dominant figure at Kusumapura and may even have been the founder of the local school. His fundamental work, the Aryabhatiya, set the agenda for research in mathematics and astronomy in India for many centuries.  
 +
 
 +
 
 +
One of Aryabhata’s discoveries was a method for solving linear equations of the form ax + by = c. Here a, b, and c are whole numbers, and we are seeking values of x and y in whole numbers satisfying the above equation. For example, if a = 5, b = 2, and c = 8, then x = 8 and y = −16 is a solution. In fact, there are infinitely many solutions:
 +
 
 
x = 8 − 2m
 
x = 8 − 2m
 
y = 5m − 16
 
y = 5m − 16
where m is any whole number, as can easily be veri-fed. Aryabhata devised a general method for solv-ing such equations, and he called it the kuṭṭaka (or pulverizer) method. He called it the pulverizer be-cause it proceeded by a series of steps, each of which required the solution of a similar problem, but with smaller numbers. Tus, a, b, and c were ‘pulverized’ into smaller numbers.
+
where m is any whole number, as can easily be verified. Aryabhata devised a general method for solving such equations, and he called it the kuṭṭaka (or pulverizer) method. He called it the pulverizer be-cause it proceeded by a series of steps, each of which required the solution of a similar problem, but with smaller numbers. Thus, a, b, and c were ‘pulverized’ into smaller numbers.
The Euclidean algorithm, which occurs in the Elements of Euclid, gives a method to compute the greatest common divisor of two numbers by a se-quence of reductions to smaller numbers. As far as I am aware, Euclid does not suggest that this method can be used to solve linear equations of the above sort. Today, it is known that if the algorithm in Euclid is applied in reverse order, then in fact it will yield Aryabhata’s method. Unfortunately, the mathematical literature still refers to this as the ex-tended Euclidean algorithm, mainly out of igno-rance of Aryabhata’s work.
+
 
It should be noted that Aryabhata studied the above linear equations because of his interest in as-tronomy. In modern times, these equations are of interest in computational number theory and are of fundamental importance in cryptography.
+
The Euclidean algorithm, which occurs in the Elements of Euclid, gives a method to compute the greatest common divisor of two numbers by a sequence of reductions to smaller numbers. As far as I am aware, Euclid does not suggest that this method can be used to solve linear equations of the above sort. Today, it is known that if the algorithm in Euclid is applied in reverse order, then in fact it will yield Aryabhata’s method. Unfortunately, the mathematical literature still refers to this as the ex-tended Euclidean algorithm, mainly out of ignorance of Aryabhata’s work.
Amongst other important contributions of Ar-yabhata is his approximation of π to four decimal places (3.1416). By comparison, the Greeks were using the weaker approximation 3.1429. Also of importance is Aryabhata’s work on trigonometry, including his tables of values of the sine function, as well as algebraic formulae for computing the sine of multiples of an angle.  
+
 
The other major centre of mathematical learn-ing during this period was Ujjain, which was home to Varahamihira, Brahmagupta, and Bhaskarach-arya. Te text Brahma-sphuta-siddhanta by Brah-magupta, published in 628 CE, dealt with arithme-tic involving zero and negative numbers.
+
 
As with Aryabhata, Brahmagupta was an as-tronomer, and much of his work was motivated by problems that arose in astronomy. He gave the fa-mous formula for a solution to the quadratic equa-tion ax2 + bx + c = 0, namely  
+
It should be noted that Aryabhata studied the above linear equations because of his interest in astronomy. In modern times, these equations are of interest in computational number theory and are of fundamental importance in cryptography.
x = −b + √b2 − 4ac
+
 
2a
+
 
It is not clear whether Brahmagupta gave just this solution or both solutions to this equation.  
+
Amongst other important contributions of Aryabhata is his approximation of π to four decimal places (3.1416). By comparison, the Greeks were using the weaker approximation 3.1429. Also of importance is Aryabhata’s work on trigonometry, including his tables of values of the sine function, as well as algebraic formulae for computing the sine of multiples of an angle.  
Brahmagupta also studied quadratic equations in two variables and sought solutions in whole num-bers. Such equations were studied only much later in Europe. We shall discuss this topic in more detail in the next section.  
+
 
This period closes with Bhaskaracharya (1200 CE). In his fundamental work on arithmetic (titled Lilavati) he refned the kuṭṭaka method of Aryab-hata and Brahmagupta. Te Lilavati is impressive for its originality and diversity of topics.  
+
 
Until recently, it was a popularly held view that there was no original Indian mathematics before Bhaskaracharya. However, the above discussion shows that his work was the culmination of a series of distinguished mathematicians who came before him. Also, afer Bhaskaracharya, there seems to have been a gap of two hundred years before the next recorded work. Perhaps this is another time period about which more research is needed.  
+
The other major centre of mathematical learning during this period was Ujjain, which was home to Varahamihira, Brahmagupta, and Bhaskaracharya. The text Brahma-sphuta-siddhanta by Brahmagupta, published in 628 CE, dealt with arithmetic involving zero and negative numbers.
The Solution of Pell’s equation
+
 
In Brahmagupta’s work, Pell’s equation had already made an appearance. Tis is the equation that, for a given whole number D, asks for whole numbers x and y satisfying the equation  
+
 
 +
As with Aryabhata, Brahmagupta was an astronomer, and much of his work was motivated by problems that arose in astronomy. He gave the famous formula for a solution to the quadratic equation ax2 + bx + c = 0, namely  
 +
x = −b + √b2 − 4ac2a
 +
 
 +
 
 +
It is not clear whether Brahmagupta gave just this solution or both solutions to this equation. Brahmagupta also studied quadratic equations in two variables and sought solutions in whole numbers. Such equations were studied only much later in Europe. We shall discuss this topic in more detail in the next section.  
 +
 
 +
 
 +
This period closes with Bhaskaracharya (1200 CE). In his fundamental work on arithmetic (titled Lilavati) he refined the kuttaka method of Aryabhata and Brahmagupta. The Lilavati is impressive for its originality and diversity of topics.  
 +
Until recently, it was a popularly held view that there was no original Indian mathematics before Bhaskaracharya. However, the above discussion shows that his work was the culmination of a series of distinguished mathematicians who came before him. Also, after Bhaskaracharya, there seems to have been a gap of two hundred years before the next recorded work. Perhaps this is another time period about which more research is needed.  
 +
 
 +
 
 +
== The Solution of Pell’s equation ==
 +
 
 +
 
 +
In Brahmagupta’s work, Pell’s equation had already made an appearance. This is the equation that, for a given whole number D, asks for whole numbers x and y satisfying the equation  
 
x2 − Dy2 = 1.
 
x2 − Dy2 = 1.
In modern times, it arises in the study of units of quadratic felds and is a topic in the feld of alge-braic number theory. If D is a whole square (such as 1, 4, 9, and so on), the equation is easy to solve, as it factors into the product(x − my)(x + my) = 1where D = m2. Tis implies that each factor is +1 or −1, and the values of x and y can be determined from that. However, if D is not a square, then it is not even clear that there is a solution. Moreover, if there is a solution, it is not clear how one can determine all so-lutions. For example, consider the case D = 2. Here, x = 3 and y = 2 gives a solution. But if D = 61, then even the smallest solutions are huge [see p. 545].
+
 
Brahmagupta discovered a method, which he called samāsa, by which, given two solutions of the equation, a third solution could be found. Tat is, he discovered a composition law on the set of solu-tions. Brahmagupta’s lemma was known one thou-sand years before it was rediscovered in Europe by Fermat, Legendre, and others.
+
 
This method appears now in most standard text-books and courses in number theory. Te name of the equation is a historical accident. Te Swiss mathematician Leonhard Euler mistakenly as-sumed that the English mathematician John Pell was the frst to formulate the equation, and began referring to it by this name.  
+
In modern times, it arises in the study of units of quadratic fields and is a topic in the field of algebraic number theory. If D is a whole square (such as 1, 4, 9, and so on), the equation is easy to solve, as it factors into the product(x − my)(x + my) = 1where D = m2. This implies that each factor is +1 or −1, and the values of x and y can be determined from that. However, if D is not a square, then it is not even clear that there is a solution. Moreover, if there is a solution, it is not clear how one can determine all solutions. For example, consider the case D = 2. Here, x = 3 and y = 2 gives a solution. But if D = 61, then even the smallest solutions are huge [see p. 545].
The work of Bhaskaracharya gives an algorith-mic approach—which he called the cakravāla (cy-clic) method—to fnding all solutions of this equa-tion. Te method depends on computing the con-tinued fraction expansion of the square root of D and using the convergents to give values of x and y. Again, this method can be found in most modern books on number theory, though the contributions of Bhaskaracharya do not seem to be well-known.
+
 
Mathematics in South India
+
 
We described above the centres at Kusumapara and Ujjain. Both of these cities are in North India. Tere was also a fourishing tradition of mathemat-ics in South India which we shall discuss in brief in this section.
+
Brahmagupta discovered a method, which he called samāsa, by which, given two solutions of the equation, a third solution could be found. That is, he discovered a composition law on the set of solutions. Brahmagupta’s lemma was known one thou-sand years before it was rediscovered in Europe by Fermat, Legendre, and others.
Mahavira is a mathematician belonging to the ninth century who was most likely from modern-day Karnataka. He studied the problem of cubic and quartic equations and solved them for some families of equations. His work had a signifcant
+
 
impact on the development of mathematics in South India. His book Ganita-sara-sangraha am-plifes the work of Brahmagupta and provides a very useful reference for the state of mathematics in his day. It is not clear what other works he may have published; further research into the extent of his contributions would probably be very fruitful.  
+
 
Another notable mathematician of South India was Madhava, from Kerala. Madhava belongs to the fourteenth century. He discovered series expan-sions for some trigonometric functions such as the sine, cosine, and arctangent that were not known in Europe until afer Newton. In modern terminology, these expansions are the Taylor series of the func-tions in question.  
+
This method appears now in most standard text-books and courses in number theory. The name of the equation is a historical accident. The Swiss mathematician Leonhard Euler mistakenly assumed that the English mathematician John Pell was the first to formulate the equation, and began referring to it by this name.  
Madhava gave an approximation to π of 3.14159265359, which goes far beyond the four dec-imal places computed by Aryabhata. Madhava de-duced his approximation from an infnite series expansion for π /4 that became known in Europe only several centuries afer Madhava (due to the work of Leibniz).  
+
 
Madhava’s work with series expansions suggests that he either discovered elements of the diferen-tial calculus or nearly did so. Tis is worth further analysis. In a work in 1835, Charles Whish suggest-ed that the Kerala school had ‘laid the foundation for a complete system of fuxions’. Te theory of fuxions is the name given by Newton to what we today call the diferential calculus. On the other hand, some scholars have been very dismissive of  
+
 
the contributions of the Kerala school, claiming that it never progressed beyond a few series expan-sions. In particular, the theory was not developed into a powerful tool as was done by Newton. We note that it was around 1498 that Vasco da Gama arrived in Kerala and the Portuguese occupation began. Judging by evidence at other sites, it is not likely that the Portuguese were interested in ei-ther encouraging or preserving the sciences of the  
+
The work of Bhaskaracharya gives an algorithmic approach—which he called the cakravāla (cyclic) method—to finding all solutions of this equation. The method depends on computing the continued fraction expansion of the square root of D and using the convergents to give values of x and y. Again, this method can be found in most modern books on number theory, though the contributions of Bhaskaracharya do not seem to be well-known.
region. No doubt, more research is needed to dis-cover where the truth lies.
+
 
Madhava spawned a school of mathematics in Kerala, and among his followers may be noted Ni-lakantha and Jyesthadeva. It is due to the writings of these mathematicians that we know abpout the work of Madhava, as all of Madhava’s own writings seem to be lost.Mathematics in the Modern Age In more recent times, there have been many impor-tant discoveries made by mathematicians of Indian origin. We shall mention the work of three of them: Srinivasa Ramanujan, Harish-Chandra, and Man-jul Bhargava.
+
 
Ramanujan (1887–1920) is perhaps the most fa-mous of modern Indian mathematicians. Tough he produced signifcant and beautiful results in many aspects of number theory, his most lasting discovery may be the arithmetic theory of modu-lar forms. In an important paper published in 1916, he initiated the study of the τ function. Te values of this function are the Fourier coefcients of the unique normalized cusp form of weight 12 for the modular group SL2(Z). Ramanujan proved some properties of the τ function and conjectured many  
+
== Mathematics in South India ==
more. As a result of his work, the modern arithme-tic theory of modular forms, which occupies a cen-tral place in number theory and algebraic geometry, was developed by Hecke.5  
+
 
Harish-Chandra (1923–83) is perhaps the least known Indian mathematician outside of mathemat-ical circles. He began his career as a physicist, work-ing under Dirac. In his thesis, he worked on the rep-resentation theory of the group SL2(C). Tis work convinced him that he was really a mathematician, and he spent the remainder of his academic life  
+
 
working on the representation theory of semi-sim-ple groups. For most of that period, he was a pro-fessor at the Institute for Advanced Study in Prin-ceton, New Jersey. His Collected Papers published in four volumes contain more than 2,000 pages.6 His style is known as meticulous and thorough and his published work tends to treat the most general case at the very outset. Tis is in contrast to many other mathematicians, whose published work tends to evolve through special cases. Interestingly, the work of Harish-Chandra formed the basis of Lang-
+
We described above the centres at Kusumapara and Ujjain. Both of these cities are in North India. There was also a flourishing tradition of mathematics in South India which we shall discuss in brief in this section.  
lands’s theory of automorphic forms, which are a vast generalization of the modular forms consid-ered by Ramanujan.7  
+
 
Manjul Bhargava (b. 1974) discovered a compo-sition  law for ternary quadratic forms. In our dis-cussion of Pell’s equation, we indicated that Brah-magupta discovered a composition law for the so-lutions. Identifying a set of importance and discov-ering an algebraic structure such as a composition law is an important theme in mathematics. Karl Gauss, one of the greatest mathematicians of all time, showed that binary quadratic forms, that is, functions of the form
+
 
 +
Mohair is a mathematician belonging to the ninth century who was most likely from modern-day Karnataka. He studied the problem of cubic and quadratic equations and solved them for some families of equations. His work had a significant impact on the development of mathematics in South India. His book Ganita-sara-sangraha amplifies the work of Brahmagupta and provides a very useful reference for the state of mathematics in his day. It is not clear what other works he may have published; further research into the extent of his contributions would probably be very fruitful.  
 +
 
 +
 
 +
Another notable mathematician of South India was Madhava, from Kerala. Madhava belongs to the fourteenth century. He discovered series expansions for some trigonometric functions such as the sine, cosine, and arctangent that were not known in Europe until after Newton. In modern terminology, these expansions are the Taylor series of the functions in question.  
 +
 
 +
 
 +
Madhava gave an approximation to π of 3.14159265359, which goes far beyond the four decimal places computed by Aryabhata. Madhava deduced his approximation from an infinite series expansion for π /4 that became known in Europe only several centuries after Madhava (due to the work of Leibniz).  
 +
 
 +
 
 +
Madhava’s work with series expansions suggests that he either discovered elements of the differential calculus or nearly did so. This is worth further analysis. In a work in 1835, Charles Whish suggested that the Kerala school had ‘laid the foundation for a complete system of fluxions’. The theory of fluxions is the name given by Newton to what we today call the differential calculus. On the other hand, some scholars have been very dismissive of the contributions of the Kerala school, claiming that it never progressed beyond a few series expansions. In particular, the theory was not developed into a powerful tool as was done by Newton. We note that it was around 1498 that Vasco da Gama arrived in Kerala and the Portuguese occupation began. Judging by evidence at other sites, it is not likely that the Portuguese were interested in either encouraging or preserving the sciences of the region. No doubt, more research is needed to discover where the truth lies.
 +
 
 +
 
 +
Madhava spawned a school of mathematics in Kerala, and among his followers may be noted Nilakantha and Jyesthadeva. It is due to the writings of these mathematicians that we know about the work of Madhava, as all of Madhava’s own writings seem to be lost.
 +
 
 +
On 22 May the King of Norway presented the Abel Prize for 2007 to the distinguished probabilist S R Srinivasa 
 +
Varadhan. Considered as the Nobel Prize for mathematics, the Abel Prize is awarded by the Norwegian Academy of
 +
Science and Letters for outstanding work in mathematics, work of extraordinary depth and influence in the 
 +
mathematical sciences. Varadhan received the prize for ‘his fundamental contributions to probability theory and in
 +
particular for creating a unified theory of large deviations’. Kristian Seip, chairman of the Abel Committee,
 +
says: ‘Varadhan’s work has great conceptual strength and ageless beauty. His ideas have been hugely influential and
 +
will continue to stimulate further research for a long time.’
 +
 
 +
 
 +
== Mathematics in the Modern Age: ==
 +
 
 +
 
 +
In more recent times, there have been many important discoveries made by mathematicians of Indian origin. We shall mention the work of three of them: Shrinivas Ramanujan, Harish-Chandra, and Manjul Bhargava.
 +
 
 +
 
 +
Ramanujan (1887–1920) is perhaps the most famous of modern Indian mathematicians. Though he produced significant and beautiful results in many aspects of number theory, his most lasting discovery may be the arithmetic theory of modular forms. In an important paper published in 1916, he initiated the study of the τ function. The values of this function are the Fourier coefficients of the unique normalized cusp form of weight 12 for the modular group SL2 (Z). Ramanujan proved some properties of the τ function and conjectured many more. As a result of his work, the modern arithmetic theory of modular forms, which occupies a central place in number theory and algebraic geometry, was developed by Hecke.5  
 +
 
 +
 
 +
Harish-Chandra (1923–83) is perhaps the least known Indian mathematician outside of mathematical circles. He began his career as a physicist, working under Dirac. In his thesis, he worked on the representation theory of the group SL2(C). This work convinced him that he was really a mathematician, and he spent the remainder of his academic life working on the representation theory of semi-simple groups. For most of that period, he was a professor at the Institute for Advanced Study in Princeton, New Jersey. His Collected Papers published in four volumes contain more than 2,000 pages.6 His style is known as meticulous and thorough and his published work tends to treat the most general case at the very outset. This is in contrast to many other mathematicians, whose published work tends to evolve through special cases. Interestingly, the work of Harish-Chandra formed the basis of Langlands’s theory of automorphic forms, which are a vast generalization of the modular forms considered by Ramanujan.7  
 +
 
 +
 
 +
Manjul Bhargava (b. 1974) discovered a composition law for ternary quadratic forms. In our discussion of Pell’s equation, we indicated that Brahmagupta discovered a composition law for the solutions. Identifying a set of importance and discovering an algebraic structure such as a composition law is an important theme in mathematics. Karl Gauss, one of the greatest mathematicians of all time, showed that binary quadratic forms, that is, functions of the form
 
ax2 + bxy + cy2
 
ax2 + bxy + cy2
where a, b, and c are integers, have such a structure. More precisely, the set of primitive SL2(Z) orbits of binary quadratic forms of a given discriminant D has the structure of an abelian group. In fact, this is the ideal class group. Afer this fundamental work of Gauss, there had been no progress for several centuries on discovering such structures in other classes of forms. Manjul Bhargava’s stunning work in his doctoral thesis, published as several papers in  
+
 
the Annals of Mathematics, shows how to address this question for cubic (and other higher degree) binary and ternary forms.8 Te work of Bhargava, who is currently Professor of Mathematics at Prin-ceton University, is deep, beautiful, and largely un-expected. It has many important ramifcations and will likely form a theme of mathematical study at least for the coming decades. It is also sure to be a topic of discussion at the 2010 International Con-gress of Mathematicians in Hyderabad.
+
where a, b, and c are integers, have such a structure. More precisely, the set of primitive SL2(Z) orbits of binary quadratic forms of a given discriminant D has the structure of an abelian group. In fact, this is the ideal class group. After this fundamental work of Gauss, there had been no progress for several centuries on discovering such structures in other classes of forms. Manjul Bhargava’s stunning work in his doctoral thesis, published as several papers in the Annals of Mathematics, shows how to address this question for cubic (and other higher degree) binary and ternary forms.8 The work of Bhargava, who is currently Professor of Mathematics at Princeton University, is deep, beautiful, and largely unexpected. It has many important ramifications and will likely form a theme of mathematical study at least for the coming decades. It is also sure to be a topic of discussion at the 2010 International Congress of Mathematicians in Hyderabad.  
Select Bibliography
+
 
1. André Weil, Number Teory: An Approach through History fom Hammurapi to Legendre (Boston: Birkhäuser, 1984).
+
 
2. George G Joseph, Te Crest of the Peacock: Non-European Roots of Mathematics (London: Penguin, 1991).
+
== Select Bibliography ==
3. Georges Ifrah, Te Universal History of Numbers: From Prehistory to the Invention of the Computer (London: Wiley , 1998).
+
 
4. A Seidenberg, ‘Te Geometry of the Vedic Rituals’,in Agni: Te Vedic Ritual of the Fire Altar, ed. Frits Staal, (Berkeley: Asian Humanities, 1983).References
+
1. André Weil, Number Theory: An Approach through History from Hammurapi to Legendre (Boston: Birkhäuser, 1984).
5. Te Complete Works of Swami Vivekananda, 9 vols (Calcutta: Advaita Ashrama, 1–8, 1989; 9, 1997), 2.20.
+
2. George G Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (London: Penguin, 1991).
6 In particular, the following papers were very use-ful in compiling this article: (i) J J O’Connor and E F Robertson, ‘An overview of Indian mathemat-ics’ <http://www-history.mcs.st- andrews.ac.uk/HistTopics/Indian_mathematics.html> accessed
+
3. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (London: Wiley , 1998).
 +
 
 +
== References ==
 +
 
 +
 
 +
4. A Seidenberg, ‘The Geometry of the Vedic Rituals’, in Agni: The Vedic Ritual of the Fire Altar, ed. Frits Staal, (Berkeley: Asian Humanities, 1983).References
 +
5. The Complete Works of Swami Vivekananda, 9 vols (Calcutta: Advaita Ashrama, 1–8, 1989; 9, 1997), 2.20.
 +
6 In particular, the following papers were very useful in compiling this article: (i) J J O’Connor and E F Robertson, ‘An overview of Indian mathematics’ <http://www-history.mcs.st- andrews.ac.uk/HistTopics/Indian_mathematics.html> accessed
 
7 July 2007; I G Pearce, ‘Indian Mathematics: Re- dressing the Balance’ <http://www-history.mcs .st-andrews.ac.uk/history/Projects/Pearce/index .html> accessed 2 July 2007.
 
7 July 2007; I G Pearce, ‘Indian Mathematics: Re- dressing the Balance’ <http://www-history.mcs .st-andrews.ac.uk/history/Projects/Pearce/index .html> accessed 2 July 2007.
 
8. See V Kumar Murty, ‘Contributions of the Indian Subcontinent to Civilization’, Prabuddha Bharata, 100/1 (1995), 134–5.
 
8. See V Kumar Murty, ‘Contributions of the Indian Subcontinent to Civilization’, Prabuddha Bharata, 100/1 (1995), 134–5.
9 Pierre Simon Laplace (1749–1827) was a French mathematician famous for, among other things, his contributions to the theory of probability and the diferential equation named afer him. 5. Ramanujan’s works are available in Collected Papers of Srinivasa Ramanujan, eds G H Hardy, P V Seshu Aiyar, and B M Wilson (Urbana: University of Il-linois, 1927).
+
9 Pierre Simon Laplace (1749–1827) was a French mathematician famous for, among other things, his contributions to the theory of probability and the differential equation named after him. 5. Ramanujan’s works are available in Collected Papers of Srinivasa Ramanujan, eds G H Hardy, P V Seshu Aiyar, and B M Wilson (Urbana: University of Illinois, 1927).
 
  10. Harish-Chandra, Collected Papers, 4 volumes, ed. V S Varadarajan (New York: Springer-Verlag, 1981).
 
  10. Harish-Chandra, Collected Papers, 4 volumes, ed. V S Varadarajan (New York: Springer-Verlag, 1981).
 
11. For a discussion, see V Kumar Murty, ‘Ramanujan and Harish- Chandra’, Mathematical Intelligencer, 15 (1993), 33–39.
 
11. For a discussion, see V Kumar Murty, ‘Ramanujan and Harish- Chandra’, Mathematical Intelligencer, 15 (1993), 33–39.
 
12 M Bhargava, ‘Higher Composition Laws’, Annals of Mathematics, 159/1 (2004), 217–50; 159/2 (2004), 865-86
 
12 M Bhargava, ‘Higher Composition Laws’, Annals of Mathematics, 159/1 (2004), 217–50; 159/2 (2004), 865-86

Revision as of 09:27, 27 April 2010

Prof. Vijaya Kumar Murty


Mathematics has played a significant role in the development of Indian culture for millennia. Mathematical ideas that originated in the Indian subcontinent have had a profound impact on the world. Swami Vivekananda said: ‘You know how many sciences had their origin in India. Mathematics began there. You are even today counting 1, 2, 3, etc. to zero, after Sanskrit figures, and you all know that algebra also originated in India.’1


It is also a fitting time to review the contributions of Indian mathematicians from ancient times to the present, as in 2010, India will be hosting the International Congress of Mathematicians. This quadrennial meeting brings together mathematicians from around the world to discuss the most significant developments in the subject over the past four years and to get a sense of where the subject is heading in the next four. The idea of holding such a congress at regular intervals actually started at the Columbian Exhibition in Chicago in 1893. This exhibition had sessions to highlight the advancement of knowledge in different fields. One of these was a session on mathematics. Another, perhaps more familiar to readers of Prabuddha Bharata, was the famous Parliament of Religions in which Swami Vivekananda first made his public appearance in the West.


Following the Chicago meeting, the first Inter-national Congress of Mathematicians took place in Zurich in 1897. It was at the next meeting at Paris in 1900 that Hilbert formulated his now famous 23 problems. Since that time, the congress has been meeting approximately every four years in different cities around the world, and in 2010, the venue will be Hyderabad, India. This is the first time in its more than hundred-year history that the congress will be held in India. This meeting could serve as an impetus and stimulus to mathematical thought in the subcontinent, pro-vided the community is prepared for it. Preparation would largely consist in being aware of the tradition of mathematics in India, from ancient times to modern, and in embracing the potential and possibility of developing this tradition to new heights in the coming millennia.


In ancient times, mathematics was mainly used in an auxiliary or applied role. Thus, mathematical methods were used to solve problems in architecture and construction (as in the public works of the Harappa civilization), in astronomy and astrology (as in the works of the Jain mathematicians), and in the construction of Vedic altars (as in the case of the Shula Sutras of Baudhayana and his successors). By the sixth or fifth century BCE, mathematics was being studied for its own sake. In modern times, mathematics is studied both for its own sake, as well as for its applications in other fields of knowledge.


The aim of this article is to give a brief review of a few of the outstanding innovations introduced by Indian mathematics from ancient times to modern. As we shall see, there does not seem to have been a time in Indian history when mathematics was not being developed. Recent work has unearthed many manuscripts, and what were previously regarded as dormant periods in Indian mathematics are now known to have been very active. Even a small study of this subject leaves one with a sense of wonder at the depth and breadth of ancient Indian thought.


The picture is not yet complete, and it seems that there is much work to do in the field of the history of Indian mathematics. The challenges are two-fold. First, there is the task of locating and identifying manuscripts and of translating them into a language that is more familiar to modern scholars. Second, there is the task of interpreting the significance of the work that was done.


Since much of the past work in this area has tended to adopt a Eurocentric perspective and interpretation, it is necessary to take a fresh, objective look. The time is ripe to make a major effort to develop as complete a picture as possible of Indian mathematics. Those who are interested in embarking on such an effort can find much helpful material online.2


We may ask what the term ‘Indian’ means in the context of this discussion. Mostly, it refers to the Indian subcontinent, but for more recent history we include also the diaspora and people whose roots can be traced to the Indian subcontinent, wherever they may be geographically located.


Mathematics in Ancient Times (3000 to 600 BCE)

The Indus valley civilization is considered to have existed around 3000 BCE. Two of its most famous cities, Harappa and Mohenjo-Daro, provide evidence that construction of buildings followed a standardized measurement which was decimal in nature. Here, we see mathematical ideas developed for the purpose of construction. This civilization had an advanced brick-making technology (having invented the kiln). Bricks were used in the construction of buildings and embankments for food control.


The study of astronomy is considered to be even older, and there must have been mathematical theories on which it was based. Even in later times, we find that astronomy motivated considerable mathematical development, especially in the field of trigonometry.


Much has been written about the mathematical constructions that are to be found in Vedic literature. In particular, the Shatapatha Brahmana, which is a part of the Shukla Yajur Veda, contains detailed descriptions of the geometric construction of altars for yajnas. Here, the brick-making technology of the Indus valley civilization was put to a new use. As usual, there are different interpretations of the dates of Vedic texts, and in the case of this Brahman, the range is from 1800 to about 800 BCE. Perhaps it is even older.


Supplementary to the Vedas are the Shula Sutras. These texts are considered to date from 800 to 200 BCE. Four in number, they are named after their authors: Baudhayana (800 BCE), Manava (750 BCE), Apastamba (600 BCE), and Katyayana (200 BCE). The sutras contain the famous theorem commonly attributed to Pythagoras. Some scholars (such as Seidenberg) feel that this theorem is already present in the Shatapatha Brahmana. Much later, Bhaskaracharya gave an algebraic proof of this theorem, as opposed to the geometric proof that the Greeks, and possibly the Chinese, were aware of.3


The Shulba Sutras introduce the concept of irrational numbers, numbers that are not the ratio of two whole numbers. For example, the square root of 2 is one such number. The sutras give a way of approximating the square root of a number using rational numbers through a recursive procedure which in modern language would be a ‘series expansion’. This predates, by far, the European use of Taylor series.


It is interesting that the mathematics of this period seems to have been developed for solving practical geometric problems, especially the construction of religious altars. However, the study of the series expansions for certain functions already hints at the development of an algebraic perspective. In later times, we find a shift towards algebra, with simplification of algebraic formulae and summation of series acting as catalysts for mathematical discovery.


Jain Mathematics (600 BCE to 500 CE)

This is a topic that scholars have started studying only recently. Knowledge of this period of mathematical history is still fragmentary, and it is a fertile area for future scholarly studies. Just as Vedic philosophy and theology stimulated the development of certain aspects of mathematics, so too did the rise of Jainism. Jain cosmology led to ideas of the infinite. This, in turn, led to the development of the notion of ‘orders of infinity’ as a mathematical concept. By ‘orders of infinity’, we mean a theory by which one set could be deemed to be ‘more infinite’ than another. In modern language, this corresponds to the notion of cardinality. For a finite set, its cardinality is the number of elements it contains. However, we need a more sophisticated notion to measure the size of an infinite set. In Europe, it was not until Cantor’s work in the nineteenth century that a proper concept of cardinality was established.


Besides the investigations into infinity, this period saw developments in several other fields such as number theory, geometry, computing with fractions, and combinatorics. In particular, the recursion formula for binomial coefficients and the ‘Pascal’s triangle’ were already known in this period


As mentioned in the previous section, astronomy had been studied in India since ancient times. This subject is often confused with astrology. Swami Vivekananda has speculated that astrology came to India from the Greeks and that astronomy was borrowed by the Greeks from India. Indirect evidence for this is provided by a text by Yavaneshvara (c. 200 CE) which popularized a Greek astrology text dating back to 120 BCE.


The period 600 BCE to 900 CE coincides with the rise and dominance of Buddhism. In the Lalitavistara, a biography of the Buddha which may have been written around the first century CE, there is an incident about Gautama being asked to state the name of large powers of 10, starting with 107. He is able to give names to numbers up to 1053 (tallaksana). The very fact that such large numbers had names suggests that the mathematicians of the day were comfortable thinking about very large numbers. It is hard to imagine calculating with such numbers without some form of place-value system.


Brahmi Numerals, the Place-value System, and Zero

No account of Indian mathematics would be complete without a discussion of Indian numerals, the place-value system, and the concept of zero. The numerals that we use even today can be traced to the Brahma numerals that seem to have made their appearance in 300 BCE. But Brahma numerals were not part of a place-value system. They evolved into the Gupta numerals around 400 CE and subsequently into the Devanagari numerals, which developed slowly between 600 and 1000 CE.


By 600 CE, a place-value decimal system was well in use in India. This means that when a number is written down, each symbol that is used has an absolute value, but also a value relative to its position. For example, the numbers 1 and 5 have a value on their own, but also have a value relative to their position in the number 15. The importance of a place-value system need hardly be emphasized. It would suffice to cite an oft-quoted remark by La-place: ‘It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.’4


A place-value system of numerals was apparently known in other cultures; for example, the Babylonians used a sexagesimal place-value system as early as 1700 BCE, but the Indian system was the first decimal system. Moreover, until 400 BCE, the Ba-bylonian system had an inherent ambiguity as there was no symbol for zero. Thus, it was not a complete place-value system in the way we think of it today.


The elevation of zero to the same status as other numbers involved difficulties that many brilliant mathematicians struggled with. The main problem was that the rules of arithmetic had to be formulated so as to include zero. While addition, subtraction, and multiplication with zero were mastered, division was a more subtle question. Today, we know that division by zero is not well-defined and so has to be excluded from the rules of arithmetic. But this understanding did not come all at once, and took the combined eforts of many minds. It is interesting to note that it was not until the seventeenth century that zero was being used in Europe, and the path of mathematics from India to Europe is the subject of much historical research.


The Classical Era of Indian Mathematics (500 to 1200 CE)

The most famous names of Indian mathematics belong to what is known as the classical era. This includes Aryabhata I (500 CE), Brahmagupta (700 CE), Bhaskara I (900 CE), Mahavira (900 CE), Aryabhata II (1000 CE), and Bhaskaracharya or Bhaskara II (1200 CE).


During this period, two centers of mathematical research emerged, one at Kusumapura near Pataliputra and the other at Ujjain. Aryabhata I was the dominant figure at Kusumapura and may even have been the founder of the local school. His fundamental work, the Aryabhatiya, set the agenda for research in mathematics and astronomy in India for many centuries.


One of Aryabhata’s discoveries was a method for solving linear equations of the form ax + by = c. Here a, b, and c are whole numbers, and we are seeking values of x and y in whole numbers satisfying the above equation. For example, if a = 5, b = 2, and c = 8, then x = 8 and y = −16 is a solution. In fact, there are infinitely many solutions:

x = 8 − 2m y = 5m − 16 where m is any whole number, as can easily be verified. Aryabhata devised a general method for solving such equations, and he called it the kuṭṭaka (or pulverizer) method. He called it the pulverizer be-cause it proceeded by a series of steps, each of which required the solution of a similar problem, but with smaller numbers. Thus, a, b, and c were ‘pulverized’ into smaller numbers.

The Euclidean algorithm, which occurs in the Elements of Euclid, gives a method to compute the greatest common divisor of two numbers by a sequence of reductions to smaller numbers. As far as I am aware, Euclid does not suggest that this method can be used to solve linear equations of the above sort. Today, it is known that if the algorithm in Euclid is applied in reverse order, then in fact it will yield Aryabhata’s method. Unfortunately, the mathematical literature still refers to this as the ex-tended Euclidean algorithm, mainly out of ignorance of Aryabhata’s work.


It should be noted that Aryabhata studied the above linear equations because of his interest in astronomy. In modern times, these equations are of interest in computational number theory and are of fundamental importance in cryptography.


Amongst other important contributions of Aryabhata is his approximation of π to four decimal places (3.1416). By comparison, the Greeks were using the weaker approximation 3.1429. Also of importance is Aryabhata’s work on trigonometry, including his tables of values of the sine function, as well as algebraic formulae for computing the sine of multiples of an angle.


The other major centre of mathematical learning during this period was Ujjain, which was home to Varahamihira, Brahmagupta, and Bhaskaracharya. The text Brahma-sphuta-siddhanta by Brahmagupta, published in 628 CE, dealt with arithmetic involving zero and negative numbers.


As with Aryabhata, Brahmagupta was an astronomer, and much of his work was motivated by problems that arose in astronomy. He gave the famous formula for a solution to the quadratic equation ax2 + bx + c = 0, namely x = −b + √b2 − 4ac2a


It is not clear whether Brahmagupta gave just this solution or both solutions to this equation. Brahmagupta also studied quadratic equations in two variables and sought solutions in whole numbers. Such equations were studied only much later in Europe. We shall discuss this topic in more detail in the next section.


This period closes with Bhaskaracharya (1200 CE). In his fundamental work on arithmetic (titled Lilavati) he refined the kuttaka method of Aryabhata and Brahmagupta. The Lilavati is impressive for its originality and diversity of topics. Until recently, it was a popularly held view that there was no original Indian mathematics before Bhaskaracharya. However, the above discussion shows that his work was the culmination of a series of distinguished mathematicians who came before him. Also, after Bhaskaracharya, there seems to have been a gap of two hundred years before the next recorded work. Perhaps this is another time period about which more research is needed.


The Solution of Pell’s equation

In Brahmagupta’s work, Pell’s equation had already made an appearance. This is the equation that, for a given whole number D, asks for whole numbers x and y satisfying the equation x2 − Dy2 = 1.


In modern times, it arises in the study of units of quadratic fields and is a topic in the field of algebraic number theory. If D is a whole square (such as 1, 4, 9, and so on), the equation is easy to solve, as it factors into the product(x − my)(x + my) = 1where D = m2. This implies that each factor is +1 or −1, and the values of x and y can be determined from that. However, if D is not a square, then it is not even clear that there is a solution. Moreover, if there is a solution, it is not clear how one can determine all solutions. For example, consider the case D = 2. Here, x = 3 and y = 2 gives a solution. But if D = 61, then even the smallest solutions are huge [see p. 545].


Brahmagupta discovered a method, which he called samāsa, by which, given two solutions of the equation, a third solution could be found. That is, he discovered a composition law on the set of solutions. Brahmagupta’s lemma was known one thou-sand years before it was rediscovered in Europe by Fermat, Legendre, and others.


This method appears now in most standard text-books and courses in number theory. The name of the equation is a historical accident. The Swiss mathematician Leonhard Euler mistakenly assumed that the English mathematician John Pell was the first to formulate the equation, and began referring to it by this name.


The work of Bhaskaracharya gives an algorithmic approach—which he called the cakravāla (cyclic) method—to finding all solutions of this equation. The method depends on computing the continued fraction expansion of the square root of D and using the convergents to give values of x and y. Again, this method can be found in most modern books on number theory, though the contributions of Bhaskaracharya do not seem to be well-known.


Mathematics in South India

We described above the centres at Kusumapara and Ujjain. Both of these cities are in North India. There was also a flourishing tradition of mathematics in South India which we shall discuss in brief in this section.


Mohair is a mathematician belonging to the ninth century who was most likely from modern-day Karnataka. He studied the problem of cubic and quadratic equations and solved them for some families of equations. His work had a significant impact on the development of mathematics in South India. His book Ganita-sara-sangraha amplifies the work of Brahmagupta and provides a very useful reference for the state of mathematics in his day. It is not clear what other works he may have published; further research into the extent of his contributions would probably be very fruitful.


Another notable mathematician of South India was Madhava, from Kerala. Madhava belongs to the fourteenth century. He discovered series expansions for some trigonometric functions such as the sine, cosine, and arctangent that were not known in Europe until after Newton. In modern terminology, these expansions are the Taylor series of the functions in question.


Madhava gave an approximation to π of 3.14159265359, which goes far beyond the four decimal places computed by Aryabhata. Madhava deduced his approximation from an infinite series expansion for π /4 that became known in Europe only several centuries after Madhava (due to the work of Leibniz).


Madhava’s work with series expansions suggests that he either discovered elements of the differential calculus or nearly did so. This is worth further analysis. In a work in 1835, Charles Whish suggested that the Kerala school had ‘laid the foundation for a complete system of fluxions’. The theory of fluxions is the name given by Newton to what we today call the differential calculus. On the other hand, some scholars have been very dismissive of the contributions of the Kerala school, claiming that it never progressed beyond a few series expansions. In particular, the theory was not developed into a powerful tool as was done by Newton. We note that it was around 1498 that Vasco da Gama arrived in Kerala and the Portuguese occupation began. Judging by evidence at other sites, it is not likely that the Portuguese were interested in either encouraging or preserving the sciences of the region. No doubt, more research is needed to discover where the truth lies.


Madhava spawned a school of mathematics in Kerala, and among his followers may be noted Nilakantha and Jyesthadeva. It is due to the writings of these mathematicians that we know about the work of Madhava, as all of Madhava’s own writings seem to be lost.

On 22 May the King of Norway presented the Abel Prize for 2007 to the distinguished probabilist S R Srinivasa  
Varadhan. Considered as the Nobel Prize for mathematics, the Abel Prize is awarded by the Norwegian Academy of 
Science and Letters for outstanding work in mathematics, work of extraordinary depth and influence in the  
mathematical sciences. Varadhan received the prize for ‘his fundamental contributions to probability theory and in 
particular for creating a unified theory of large deviations’. Kristian Seip, chairman of the Abel Committee, 
says: ‘Varadhan’s work has great conceptual strength and ageless beauty. His ideas have been hugely influential and 
will continue to stimulate further research for a long time.’


Mathematics in the Modern Age:

In more recent times, there have been many important discoveries made by mathematicians of Indian origin. We shall mention the work of three of them: Shrinivas Ramanujan, Harish-Chandra, and Manjul Bhargava.


Ramanujan (1887–1920) is perhaps the most famous of modern Indian mathematicians. Though he produced significant and beautiful results in many aspects of number theory, his most lasting discovery may be the arithmetic theory of modular forms. In an important paper published in 1916, he initiated the study of the τ function. The values of this function are the Fourier coefficients of the unique normalized cusp form of weight 12 for the modular group SL2 (Z). Ramanujan proved some properties of the τ function and conjectured many more. As a result of his work, the modern arithmetic theory of modular forms, which occupies a central place in number theory and algebraic geometry, was developed by Hecke.5


Harish-Chandra (1923–83) is perhaps the least known Indian mathematician outside of mathematical circles. He began his career as a physicist, working under Dirac. In his thesis, he worked on the representation theory of the group SL2(C). This work convinced him that he was really a mathematician, and he spent the remainder of his academic life working on the representation theory of semi-simple groups. For most of that period, he was a professor at the Institute for Advanced Study in Princeton, New Jersey. His Collected Papers published in four volumes contain more than 2,000 pages.6 His style is known as meticulous and thorough and his published work tends to treat the most general case at the very outset. This is in contrast to many other mathematicians, whose published work tends to evolve through special cases. Interestingly, the work of Harish-Chandra formed the basis of Langlands’s theory of automorphic forms, which are a vast generalization of the modular forms considered by Ramanujan.7


Manjul Bhargava (b. 1974) discovered a composition law for ternary quadratic forms. In our discussion of Pell’s equation, we indicated that Brahmagupta discovered a composition law for the solutions. Identifying a set of importance and discovering an algebraic structure such as a composition law is an important theme in mathematics. Karl Gauss, one of the greatest mathematicians of all time, showed that binary quadratic forms, that is, functions of the form ax2 + bxy + cy2

where a, b, and c are integers, have such a structure. More precisely, the set of primitive SL2(Z) orbits of binary quadratic forms of a given discriminant D has the structure of an abelian group. In fact, this is the ideal class group. After this fundamental work of Gauss, there had been no progress for several centuries on discovering such structures in other classes of forms. Manjul Bhargava’s stunning work in his doctoral thesis, published as several papers in the Annals of Mathematics, shows how to address this question for cubic (and other higher degree) binary and ternary forms.8 The work of Bhargava, who is currently Professor of Mathematics at Princeton University, is deep, beautiful, and largely unexpected. It has many important ramifications and will likely form a theme of mathematical study at least for the coming decades. It is also sure to be a topic of discussion at the 2010 International Congress of Mathematicians in Hyderabad.


Select Bibliography

1. André Weil, Number Theory: An Approach through History from Hammurapi to Legendre (Boston: Birkhäuser, 1984). 2. George G Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (London: Penguin, 1991). 3. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer (London: Wiley , 1998).

References

4. A Seidenberg, ‘The Geometry of the Vedic Rituals’, in Agni: The Vedic Ritual of the Fire Altar, ed. Frits Staal, (Berkeley: Asian Humanities, 1983).References 5. The Complete Works of Swami Vivekananda, 9 vols (Calcutta: Advaita Ashrama, 1–8, 1989; 9, 1997), 2.20. 6 In particular, the following papers were very useful in compiling this article: (i) J J O’Connor and E F Robertson, ‘An overview of Indian mathematics’ <http://www-history.mcs.st- andrews.ac.uk/HistTopics/Indian_mathematics.html> accessed 7 July 2007; I G Pearce, ‘Indian Mathematics: Re- dressing the Balance’ <http://www-history.mcs .st-andrews.ac.uk/history/Projects/Pearce/index .html> accessed 2 July 2007. 8. See V Kumar Murty, ‘Contributions of the Indian Subcontinent to Civilization’, Prabuddha Bharata, 100/1 (1995), 134–5. 9 Pierre Simon Laplace (1749–1827) was a French mathematician famous for, among other things, his contributions to the theory of probability and the differential equation named after him. 5. Ramanujan’s works are available in Collected Papers of Srinivasa Ramanujan, eds G H Hardy, P V Seshu Aiyar, and B M Wilson (Urbana: University of Illinois, 1927).

10.	 Harish-Chandra,	Collected Papers,	4	volumes,	ed.	V S	Varadarajan	(New	York:	Springer-Verlag,	1981).

11. For a discussion, see V Kumar Murty, ‘Ramanujan and Harish- Chandra’, Mathematical Intelligencer, 15 (1993), 33–39. 12 M Bhargava, ‘Higher Composition Laws’, Annals of Mathematics, 159/1 (2004), 217–50; 159/2 (2004), 865-86