# An overview of Indian mathematics

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

We shall look briefly at the Indian development of the place-value decimal system of numbers later in this article and in somewhat more detail in the separate article Indian numerals. First, however, we go back to the first evidence of mathematics developing in India.Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the *Shatapatha Brahmana* and the *Taittiriya Samhita.* Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.

The first mathematics which we shall describe in this article developed in the Indus valley. The earliest known urban Indian culture was first identified in 1921 at Harappa in the Punjab and then, one year later, at Mohenjo-daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by our term “Indian mathematics” which, in this article, refers to mathematics developed in the Indian subcontinent. The Indus civilisation (or Harappan civilisation as it is sometimes known) was based in these two cities and also in over a hundred small towns and villages. It was a civilisation which began around 2500 BC and survived until 1700 BC or later. The people were literate and used a written script containing around 500 characters which some have claimed to have deciphered but, being far from clear that this is the case, much research remains to be done before a full appreciation of the mathematical achievements of this ancient civilisation can be fully assessed.

We often think of Egyptians and Babylonians as being the height of civilisation and of mathematical skills around the period of the Indus civilisation, yet V G Childe in *New Light on the Most Ancient East* (1952) wrote:-

India confronts Egypt and Babylonia by the3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture.

We do know that the Harappans had adopted a uniform system of weights and measures. An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several scales for the measurement of length were also discovered during excavations. One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the “Indus inch”. Of course ten units is then 13.2 inches which is quite believable as the measure of a “foot”. A similar measure based on the length of a foot is present in other parts of Asia and beyond. Another scale was discovered when a bronze rod was found which was marked in lengths of 0.367 inches. It is certainly surprising the accuracy with which these scales are marked. Now 100 units of this measure is 36.7 inches which is the measure of a stride. Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in construction.It is unclear exactly what caused the decline in the Harappan civilisation. Historians have suggested four possible causes: a change in climatic patterns and a consequent agricultural crisis; a climatic disaster such flooding or severe drought; disease spread by epidemic; or the invasion of Indo-Aryans peoples from the north. The favourite theory used to be the last of the four, but recent opinions favour one of the first three. What is certainly true is that eventually the Indo-Aryans peoples from the north did spread over the region. This brings us to the earliest literary record of Indian culture, the Vedas which were composed in Vedic Sanskrit, between 1500 BC and 800 BC. At first these texts, consisting of hymns, spells, and ritual observations, were transmitted orally. Later the texts became written works for use of those practicing the Vedic religion.

The next mathematics of importance on the Indian subcontinent was associated with these religious texts. It consisted of the Sulbasutras which were appendices to the Vedas giving rules for constructing altars. They contained quite an amount of geometrical knowledge, but the mathematics was being developed, not for its own sake, but purely for practical religious purposes. The mathematics contained in the these texts is studied in some detail in the separate article on the Sulbasutras.

The main Sulbasutras were composed by Baudhayana (about 800 BC), Manava (about 750 BC), Apastamba (about 600 BC), and Katyayana (about 200 BC). These men were both priests and scholars but they were not mathematicians in the modern sense. Although we have no information on these men other than the texts they wrote, we have included them in our biographies of mathematicians. There is another scholar, who again was not a mathematician in the usual sense, who lived around this period. That was Panini who achieved remarkable results in his studies of Sanskrit grammar. Now one might reasonably ask what Sanskrit grammar has to do with mathematics. It certainly has something to do with modern theoretical computer science, for a mathematician or computer scientist working with formal language theory will recognise just how modern some of Panini‘s ideas are.

Before the end of the period of the Sulbasutras, around the middle of the third century BC, the Brahmi numerals had begun to appear.

Here is **one style of the Brahmi numerals**..

These are the earliest numerals which, after a multitude of changes, eventually developed into the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 used today. The development of numerals and place-valued number systems are studied in the article Indian numerals.

The Vedic religion with its sacrificial rites began to wane and other religions began to replace it. One of these was Jainism, a religion and philosophy which was founded in India around the 6^{th}century BC. Although the period after the decline of the Vedic religion up to the time of Aryabhata I around 500 AD used to be considered as a dark period in Indian mathematics, recently it has been recognised as a time when many mathematical ideas were considered. In fact Aryabhata is now thought of as summarising the mathematical developments of the Jaina as well as beginning the next phase.

The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. More surprisingly the Jaina developed a theory of the infinite containing different levels of infinity, a primitive understanding of indices, and some notion of logarithms to base 2. One of the difficult problems facing historians of mathematics is deciding on the date of the Bakhshali manuscript. If this is a work which is indeed from 400 AD, or at any rate a copy of a work which was originally written at this time, then our understanding of the achievements of Jaina mathematics will be greatly enhanced. While there is so much uncertainty over the date, a topic discussed fully in our article on the Bakhshali manuscript, then we should avoid rewriting the history of the Jaina period in the light of the mathematics contained in this remarkable document.

You can see a separate article about Jaina mathematics.

If the Vedic religion gave rise to a study of mathematics for constructing sacrificial altars, then it was Jaina cosmology which led to ideas of the infinite in Jaina mathematics. Later mathematical advances were often driven by the study of astronomy. Well perhaps it would be more accurate to say that astrology formed the driving force since it was that “science” which required accurate information about the planets and other heavenly bodies and so encouraged the development of mathematics. Religion too played a major role in astronomical investigations in India for accurate calendars had to be prepared to allow religious observances to occur at the correct times. Mathematics then was still an applied science in India for many centuries with mathematicians developing methods to solve practical problems.

Yavanesvara, in the second century AD, played an important role in popularising astrology when he translated a Greek astrology text dating from 120 BC. If he had made a literal translation it is doubtful whether it would have been of interest to more than a few academically minded people. He popularised the text, however, by resetting the whole work into Indian culture using Hindu images with the Indian caste system integrated into his text.

By about 500 AD the classical era of Indian mathematics began with the work of Aryabhata. His work was both a summary of Jaina mathematics and the beginning of new era for astronomy and mathematics. His ideas of astronomy were truly remarkable. He replaced the two demons Rahu, the Dhruva Rahu which causes the phases of the Moon and the Parva Rahu which causes an eclipse by covering the Moon or Sun or their light, with a modern theory of eclipses. He introduced trigonometry in order to make his astronomical calculations, based on the Greek epicycle theory, and he solved with integer solutions indeterminate equations which arose in astronomical theories.

Aryabhata headed a research centre for mathematics and astronomy at Kusumapura in the northeast of the Indian subcontinent. There a school studying his ideas grew up there but more than that, Aryabhata set the agenda for mathematical and astronomical research in India for many centuries to come. Another mathematical and astronomical centre was at Ujjain, also in the north of the Indian subcontinent, which grew up around the same time as Kusumapura. The most important of the mathematicians at this second centre was Varahamihira who also made important contributions to astronomy and trigonometry.

The main ideas of Jaina mathematics, particularly those relating to its cosmology with its passion for large finite numbers and infinite numbers, continued to flourish with scholars such as Yativrsabha. He was a contemporary of Varahamihira and of the slightly older Aryabhata. We should also note that the two schools at Kusumapura and Ujjain were involved in the continuing developments of the numerals and of place-valued number systems. The next figure of major importance at the Ujjain school was Brahmagupta near the beginning of the seventh century AD and he would make one of the most major contributions to the development of the numbers systems with his remarkable contributions on negative numbers and zero. It is a sobering thought that eight hundred years later European mathematics would be struggling to cope without the use of negative numbers and of zero.

These were certainly not Brahmagupta‘s only contributions to mathematics. Far from it for he made other major contributions in to the understanding of integer solutions to indeterminate equations and to interpolation formulas invented to aid the computation of sine tables.

The way that the contributions of these mathematicians were prompted by a study of methods in spherical astronomy is described in [25]:-

The Hindu astronomers did not possess a general method for solving problems in spherical astronomy, unlike the Greeks who systematically followed the method of Ptolemy, based on the well-known theorem of Menelaus. But, by means of suitable constructions within the armillary sphere, they were able to reduce many of their problems to comparison of similar right-angled plane triangles. In addition to this device, they sometimes also used the theory of quadratic equations, or applied the method of successive approximations. … Of the methods taught by Aryabhata and demonstrated by his scholiast Bhaskara I, some are based on comparison of similar right-angled plane triangles, and others are derived from inference. Brahmagupta is probably the earliest astronomer to have employed the theory of quadratic equations and the method of successive approximations to solving problems in spherical astronomy.

Before continuing to describe the developments through the classical period we should explain the mechanisms which allowed mathematics to flourish in India during these centuries. The educational system in India at this time did not allow talented people with ability to receive training in mathematics or astronomy. Rather the whole educational system was family based. There were a number of families who carried the traditions of astrology, astronomy and mathematics forward by educating each new generation of the family in the skills which had been developed. We should also note that astronomy and mathematics developed on their own, separate for the development of other areas of knowledge.Now a “mathematical family” would have a library which contained the writing of the previous generations. These writings would most likely be commentaries on earlier works such as the *Aryabhatiya* of Aryabhata. Many of the commentaries would be commentaries on commentaries on commentaries etc. Mathematicians often wrote commentaries on their own work. They would not be aiming to provide texts to be used in educating people outside the family, nor would they be looking for innovative ideas in astronomy. Again religion was the key, for astronomy was considered to be of divine origin and each family would remain faithful to the revelations of the subject as presented by their gods. To seek fundamental changes would be unthinkable for in asking others to accept such changes would be essentially asking them to change religious belief. Nor do these men appear to have made astronomical observations in any systematic way. Some of the texts do claim that the computed data presented in them is in better agreement with observation than that of their predecessors but, despite this, there does not seem to have been a major observational programme set up. Paramesvara in the late fourteenth century appears to be one of the first Indian mathematicians to make systematic observations over many years.

Mathematics however was in a different position. It was only a tool used for making astronomical calculations. If one could produce innovative mathematical ideas then one could exhibit the truths of astronomy more easily. The mathematics therefore had to lead to the same answers as had been reached before but it was certainly good if it could achieve these more easily or with greater clarity. This meant that despite mathematics only being used as a computational tool for astronomy, the brilliant Indian scholars were encouraged by their culture to put their genius into advances in this topic.

A contemporary of Brahmagupta who headed the research centre at Ujjain was Bhaskara I who led the Asmaka school. This school would have the study of the works of Aryabhata as their main concern and certainly Bhaskara was commentator on the mathematics of Aryabhata. More than 100 years after Bhaskara lived the astronomer Lalla, another commentator on Aryabhata.

The ninth century saw mathematical progress with scholars such as Govindasvami, Mahavira, Prthudakasvami, Sankara, and Sridhara. Some of these such as Govindasvami and Sankara were commentators on the text of Bhaskara I while Mahavira was famed for his updating of Brahmagupta‘s book. This period saw developments in sine tables, solving equations, algebraic notation, quadratics, indeterminate equations, and improvements to the number systems. The agenda was still basically that set by Aryabhata and the topics being developed those in his work.

The main mathematicians of the tenth century in India were Aryabhata II and Vijayanandi, both adding to the understanding of sine tables and trigonometry to support their astronomical calculations. In the eleventh century Sripati and Brahmadeva were major figures but perhaps the most outstanding of all was Bhaskara II in the twelfth century. He worked on algebra, number systems, and astronomy. He wrote beautiful texts illustrated with mathematical problems, some of which we present in his biography, and he provided the best summary of the mathematics and astronomy of the classical period.

Bhaskara II may be considered the high point of Indian mathematics but at one time this was all that was known [26]:-

*For a long time Western scholars thought that Indians had not done any original work till the time of Bhaskara II. This is far from the truth. Nor has the growth of Indian mathematics stopped with Bhaskara II. Quite a few results of Indian mathematicians have been rediscovered by Europeans. For instance, the development of number theory, the theory of indeterminates infinite series expressions for sine, cosine and tangent, computational mathematics, etc.*

Following Bhaskara II there was over 200 years before any other major contributions to mathematics were made on the Indian subcontinent. In fact for a long time it was thought that Bhaskara II represented the end of mathematical developments in the Indian subcontinent until modern times. However in the second half of the fourteenth century Mahendra Suri wrote the first Indian treatise on the astrolabe and Narayana wrote an important commentary on Bhaskara II, making important contributions to algebra and magic squares. The most remarkable contribution from this period, however, was by Madhava who invented Taylor series and rigorous mathematical analysis in some inspired contributions. Madhava was from Kerala and his work there inspired a school of followers such as Nilakantha and Jyesthadeva.Some of the remarkable discoveries of the Kerala mathematicians are described in [26]. These include: a formula for the ecliptic; the Newton–Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier‘s formula for the circumradius of a cyclic quadrilateral. Of particular interest is the approximation to the value of π which was the first to be made using a series. Madhava‘s result which gave a series for π, translated into the language of modern mathematics, reads

*R*= 4

*R*– 4

*R*/3 + 4

*R*/5 – …

This formula, as well as several others referred to above, were rediscovered by European mathematicians several centuries later. Madhava also gave other formulae for π, one of which leads to the approximation 3.14159265359.The first person in modern times to realise that the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus by nearly 300 years was Charles Whish in 1835. Whish‘s publication in the *Transactions of the Royal Asiatic Society of Great Britain and Ireland* was essentially unnoticed by historians of mathematics. Only 100 years later in the 1940s did historians of mathematics look in detail at the works of Kerala’s mathematicians and find that the remarkable claims made by Whish were essentially true. See for example [15]. Indeed the Kerala mathematicians had, as Whish wrote:-

… laid the foundation for a complete system of fluxions …

and these works:-

… abound with fluxional forms and series to be found in no work of foreign countries.

There were other major advances in Kerala at around this time. Citrabhanu was a sixteenth century mathematicians from Kerala who gave integer solutions to twenty-one types of systems of two algebraic equations. These types are all the possible pairs of equations of the following seven forms:

*x*+

*y*=

*a*,

*x*–

*y*=

*b*,

*xy*=

*c*,

*x*

^{2}+

*y*

^{2}=

*d*,

*x*

^{2}–

*y*

^{2}=

*e*,

*x*

^{3}+

*y*

^{3}=

*f*, and

*x*

^{3}–

*y*

^{3}=

*g*.

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. See [12] for more details.Now we have presented the latter part of the history of Indian mathematics in an unlikely way. That there would be essentially no progress between the contributions of Bhaskara II and the innovations of Madhava, who was far more innovative than any other Indian mathematician producing a totally new perspective on mathematics, seems unlikely. Much more likely is that we are unaware of the contributions made over this 200 year period which must have provided the foundations on which Madhava built his theories.

Our understanding of the contributions of Indian mathematicians has changed markedly over the last few decades. Much more work needs to be done to further our understanding of the contributions of mathematicians whose work has sadly been lost, or perhaps even worse, been ignored. Indeed work is now being undertaken and we should soon have a better understanding of this important part of the history of mathematics.

# Indian numerals

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

The purpose of this article is to attempt the difficult task of trying to describe how the Indians developed this ingenious system. We will examine two different aspects of the Indian number systems in this article. First we will examine the way that the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 evolved into the form which we recognise today. Of course it is important to realise that there is still no standard way of writing these numerals. The different fonts on this computer can produce many forms of these numerals which, although recognisable, differ markedly from each other. Many hand-written versions are even hard to recognise.The second aspect of the Indian number system which we want to investigate here is the place value system which, as Laplace comments in the quote which we gave at the beginning of this article, seems “so simple that its significance and profound importance is no longer appreciated.” We should also note the fact, which is important to both aspects, that the Indian number systems are almost exclusively base 10, as opposed to the Babylonian base 60 systems.

Beginning with the numerals themselves, we certainly know that today’s symbols took on forms close to that which they presently have in Europe in the 15^{th} century. It was the advent of printing which motivated the standardisation of the symbols. However we must not forget that many countries use symbols today which are quite different from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and unless one learns these symbols they are totally unrecognisable as for example the Greek alphabet is to someone unfamiliar with it.

One of the important sources of information which we have about Indian numerals comes from al-Biruni. During the 1020s al-Biruni made several visits to India. Before he went there al-Biruni already knew of Indian astronomy and mathematics from Arabic translations of some Sanskrit texts. In India he made a detailed study of Hindu philosophy and he also studied several branches of Indian science and mathematics. Al-Biruni wrote 27 works on India and on different areas of the Indian sciences. In particular his account of Indian astronomy and mathematics is a valuable contribution to the study of the history of Indian science. Referring to the Indian numerals in a famous book written about 1030 he wrote:-

Whilst we use letters for calculation according to their numerical value, the Indians do not use letters at all for arithmetic. And just as the shape of the letters that they use for writing is different in different regions of their country, so the numerical symbols vary.

It is reasonable to ask where the various symbols for numerals which al-Biruni saw originated. Historians trace them all back to the Brahmi numerals which came into being around the middle of the third century BC. Now these Brahmi numerals were not just symbols for the numbers between 1 and 9. The situation is much more complicated for it was not a place-value system so there were symbols for many more numbers. Also there were no special symbols for 2 and 3, both numbers being constructed from the symbol for 1.

Here is the **Brahmi one, two, three**.

There were separate Brahmi symbols for 4, 5, 6, 7, 8, 9 but there were also symbols for 10, 100, 1000, … as well as 20, 30, 40, … , 90 and 200, 300, 400, …, 900.

The Brahmi numerals have been found in inscriptions in caves and on coins in regions near Poona, Bombay, and Uttar Pradesh. Dating these numerals tells us that they were in use over quite a long time span up to the 4^{th} century AD. Of course different inscriptions differ somewhat in the style of the symbols.

Here is **one style of the Brahmi numerals**.

We should now look both forward and backward from the appearance of the Brahmi numerals. Moving forward leads to many different forms of numerals but we shall choose to examine only the path which has led to our present day symbols. First, however, we look at a number of different theories concerning the origin of the Brahmi numerals.

There is no problem in understanding the symbols for 1, 2, and 3. However the symbols for 4, … , 9 appear to us to have no obvious link to the numbers they represent. There have been quite a number of theories put forward by historians over many years as to the origin of these numerals. In [1] Ifrah lists a number of the hypotheses which have been put forward.

- The Brahmi numerals came from the Indus valley culture of around 2000 BC.
- The Brahmi numerals came from Aramaean numerals.
- The Brahmi numerals came from the Karoshthi alphabet.
- The Brahmi numerals came from the Brahmi alphabet.
- The Brahmi numerals came from an earlier alphabetic numeral system, possibly due to Panini.
- The Brahmi numerals came from Egypt.

Basically these hypotheses are of two types. One is that the numerals came from an alphabet in a similar way to the Greek numerals which were the initial letters of the names of the numbers. The second type of hypothesis is that they derive from an earlier number system of the same broad type as Roman numerals. For example the Aramaean numerals of hypothesis 2 are based on I (one) and X (four):

Ifrah examines each of the six hypotheses in turn and rejects them, although one would have to say that in some cases it is more due to lack of positive evidence rather than to negative evidence.Ifrah proposes a theory of his own in [1], namely that:-

… the first nine Brahmi numerals constituted the vestiges of an old indigenous numerical notation, where the nine numerals were represented by the corresponding number of vertical lines … To enable the numerals to be written rapidly, in order to save time, these groups of lines evolved in much the same manner as those of old Egyptian Pharonic numerals. Taking into account the kind of material that was written on in India over the centuries(tree bark or palm leaves)and the limitations of the tools used for writing(calamus or brush), the shape of the numerals became more and more complicated with the numerous ligatures, until the numerals no longer bore any resemblance to the original prototypes.

It is a nice theory, and indeed could be true, but there seems to be absolutely no positive evidence in its favour. The idea is that they evolved from:

One might hope for evidence such as discovering numerals somewhere on this evolutionary path. However, it would appear that we will never find convincing proof for the origin of the Brahmi numerals.

If we examine the route which led from the Brahmi numerals to our present symbols (and ignore the many other systems which evolved from the Brahmi numerals) then we next come to the Gupta symbols. The Gupta period is that during which the Gupta dynasty ruled over the Magadha state in northeastern India, and this was from the early 4^{th} century AD to the late 6^{th} century AD. The Gupta numerals developed from the Brahmi numerals and were spread over large areas by the Gupta empire as they conquered territory.

The Gupta numerals evolved into the Nagari numerals, sometimes called the Devanagari numerals. This form evolved from the Gupta numerals beginning around the 7^{th} century AD and continued to develop from the 11^{th} century onward. The name literally means the “writing of the gods” and it was the considered the most beautiful of all the forms which evolved. For example al-Biruni writes:-

What we[the Arabs]use for numerals is a selection of the best and most regular figures in India.

These “most regular figures” which al-Biruni refers to are the Nagari numerals which had, by his time, been transmitted into the Arab world. The way in which the Indian numerals were spread to the rest of the world between the 7^{th} to the 16^{th} centuries in examined in detail in [7]. In this paper, however, Gupta claims that Indian numerals had reached Southern Europe by the end of the 5^{th} century but his argument is based on the *Geometry* of Boethius which is now known to be a forgery dating from the first half of the 11^{th} century. It would appear extremely unlikely that the Indian numerals reach Europe as early as Gupta suggests.We now turn to the second aspect of the Indian number system which we want to examine in this article, namely the fact that it was a place-value system with the numerals standing for different values depending on their position relative to the other numerals. Although our place-value system is a direct descendant of the Indian system, we should note straight away that the Indians were not the first to develop such a system. The Babylonians had a place-value system as early as the 19^{th} century BC but the Babylonian systems were to base 60. The Indians were the first to develop a base 10 positional system and, considering the date of the Babylonian system, it came very late indeed.

The oldest dated Indian document which contains a number written in the place-value form used today is a legal document dated 346 in the Chhedi calendar which translates to a date in our calendar of 594 AD. This document is a donation charter of Dadda III of Sankheda in the Bharukachcha region. The only problem with it is that some historians claim that the date has been added as a later forgery. Although it was not unusual for such charters to be modified at a later date so that the property to which they referred could be claimed by someone who was not the rightful owner, there seems no conceivable reason to forge the date on this document. Therefore, despite the doubts, we can be fairly sure that this document provides evidence that a place-value system was in use in India by the end of the 6^{th} century.

Many other charters have been found which are dated and use of the place-value system for either the date or some other numbers within the text. These include:

- a donation charter of Dhiniki dated 794 in the Vikrama calendar which translates to a date in our calendar of 737 AD.
- an inscription of Devendravarman dated 675 in the Shaka calendar which translates to a date in our calendar of 753 AD.
- a donation charter of Danidurga dated 675 in the Shaka calendar which translates to a date in our calendar of 737 AD.
- a donation charter of Shankaragana dated 715 in the Shaka calendar which translates to a date in our calendar of 793 AD.
- a donation charter of Nagbhata dated 872 in the Vikrama calendar which translates to a date in our calendar of 815 AD.
- an inscription of Bauka dated 894 in the Vikrama calendar which translates to a date in our calendar of 837 AD.

All of these are claimed to be forgeries by some historians but some, or all, may well be genuine.The first inscription which is dated and is not disputed is the inscription at Gwalior dated 933 in the Vikrama calendar which translates to a date in our calendar of 876 AD. Further details of this inscription is given in the article on zero.

There is indirect evidence that the Indians developed a positional number system as early as the first century AD. The evidence is found from inscriptions which, although not in India, have been found in countries which were assimilating Indian culture. Another source is the Bakhshali manuscript which contains numbers written in place-value notation. The problem here is the dating of this manuscript, a topic which is examined in detail in our article on the Bakhshali manuscript.

We are left, of course, with asking the question of why the Indians developed such an ingenious number system when the ancient Greeks, for example, did not. A number of theories have been put forward concerning this question. Some historians believe that the Babylonian base 60 place-value system was transmitted to the Indians via the Greeks. We have commented in the article on zero about Greek astronomers using the Babylonian base 60 place-value system with a symbol o similar to our zero. The theory here is that these ideas were transmitted to the Indians who then combined this with their own base 10 number systems which had existed in India for a very long time.

A second hypothesis is that the idea for place-value in Indian number systems came from the Chinese. In particular the Chinese had pseudo-positional number rods which, it is claimed by some, became the basis of the Indian positional system. This view is put forward by, for example, Lay Yong Lam; see for example [8]. Lam argues that the Chinese system already contained what he calls the:-

… three essential features of our numeral notation system:(i)nine signs and the concept of zero,(ii)a place value system and(iii)a decimal base.

A third hypothesis is put forward by Joseph in [2]. His idea is that the place-value in Indian number systems is something which was developed entirely by the Indians. He has an interesting theory as to why the Indians might be pushed into such an idea. The reason, Joseph believes, is due to the Indian fascination with large numbers. Freudenthal is another historian of mathematics who supports the theory that the idea came entirely from within India.To see clearly this early Indian fascination with large numbers, we can take a look at the *Lalitavistara* which is an account of the life of Gautama Buddha. It is hard to date this work since it underwent continuous development over a long period but dating it to around the first or second century AD is reasonable. In *Lalitavistara* Gautama, when he is a young man, is examined on mathematics. He is asked to name all the numerical ranks beyond a koti which is 10^{7}. He lists the powers of 10 up to 10^{53}. Taking this as a first level he then carries on to a second level and gets eventually to 10^{421}. Gautama’s examiner says:-

You, not I, are the master mathematician.

It is stories such as this, and many similar ones, which convince Joseph that the fascination of the Indians with large numbers must have driven them to invent a system in which such numbers are easily expressed, namely a place-valued notation. He writes in [2]:-

The early use of such large numbers eventually led to the adoption of a series of names for successive powers of ten. The importance of these number names cannot be exaggerated. The word-numeral system, later replaced by an alphabetic notation, was the logical outcome of proceeding by multiples of ten. … The decimal place-value system developed when a decimal scale came to be associated with the value of the places of the numbers arranged left to right or right to left. and this was precisely what happened in India …

However, the same story in *Lalitavistara* convinces Kaplan (see [3]) that the Indians’ ideas of numbers came from the Greeks, for to him the story is an Indian version of Archimedes‘ *Sand-reckoner.* All that we know is that the place-value system of the Indians, however it arose, was transmitted to the Arabs and later into Europe to have, in the words of Laplace, *profound importance* on the development of mathematics.

# The Indian Sulbasutras

^{th}to the 5

^{th}century BC and were used for sacrificial rites which were the main feature of the religion. There was a ritual which took place at an altar where food, also sometimes animals, were sacrificed. The Vedas contain recitations and chants to be used at these ceremonies. Later prose was added called Brahmanas which explained how the texts were to be used in the ceremonies. They also tell of the origin and the importance of the sacrificial rites themselves.The Sulbasutras are appendices to the Vedas which give rules for constructing altars. If the ritual sacrifice was to be successful then the altar had to conform to very precise measurements. The people made sacrifices to their gods so that the gods might be pleased and give the people plenty food, good fortune, good health, long life, and lots of other material benefits. For the gods to be pleased everything had to be carried out with a very precise formula, so mathematical accuracy was seen to be of the utmost importance. We should also note that there were two types of sacrificial rites, one being a large public gathering while the other was a small family affair. Different types of altars were necessary for the two different types of ceremony.

All that is known of Vedic mathematics is contained in the Sulbasutras. This in itself gives us a problem, for we do not know if these people undertook mathematical investigations for their own sake, as for example the ancient Greeks did, or whether they only studied mathematics to solve problems necessary for their religious rites. Some historians have argued that mathematics, in particular geometry, must have also existed to support astronomical work being undertaken around the same period.

Certainly the Sulbasutras do not contain any proofs of the rules which they describe. Some of the rules, such as the method of constructing a square of area equal to a given rectangle, are exact. Others, such as constructing a square of area equal to that of a given circle, are approximations. We shall look at both of these examples below but the point we wish to make here is that the Sulbasutras make no distinction between the two. Did the writers of the Sulbasutras know which methods were exact and which were approximations?

The Sulbasutras were written by a scribe, although he was not the type of scribe who merely makes a copy of an existing document but one who put in considerable content and all the mathematical results may have been due to these scribes. We know nothing of the men who wrote the Sulbasutras other than their names and a rough indication of the period in which they lived. Like many ancient mathematicians our only knowledge of them is their writings. The most important of these documents are the Baudhayana Sulbasutra written about 800 BC and the Apastamba Sulbasutra written about 600 BC. Historians of mathematics have also studied and written about other Sulbasutras of lesser importance such as the Manava Sulbasutra written about 750 BC and the Katyayana Sulbasutra written about 200 BC.

Let us now examine some of the mathematics contained within the Sulbasutras. The first result which was clearly known to the authors is Pythagoras‘s theorem. The Baudhayana Sulbasutra gives only a special case of the theorem explicitly:-

The rope which is stretched across the diagonal of a square produces an area double the size of the original square.

The Katyayana Sulbasutra however, gives a more general version:-

The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together.

The diagram on the right illustrates this result.

Note here that the results are stated in terms of “ropes”. In fact, although sulbasutras originally meant rules governing religious rites, sutras came to mean a rope for measuring an altar. While thinking of explicit statements of Pythagoras‘s theorem, we should note that as it is used frequently there are many examples of Pythagorean triples in the Sulbasutras. For example (5, 12, 13), (12, 16, 20), (8, 15, 17), (15, 20, 25), (12, 35, 37), (15, 36, 39), (^{5}/_{2} , 6, ^{13}/_{2}), and (^{15}/_{2} , 10, ^{25}/_{2}) all occur.

Now the Sulbasutras are really construction manuals for geometric shapes such as squares, circles, rectangles, etc. and we illustrate this with some examples.

The first construction we examine occurs in most of the different Sulbasutras. It is a construction, based on Pythagoras‘s theorem, for making a square equal in area to two given unequal squares.

Consider the diagram on the right.

*ABCD* and *PQRS* are the two given squares. Mark a point *X* on *PQ* so that *PX* is equal to *AB*. Then the square on *SX* has area equal to the sum of the areas of the squares *ABCD* and *PQRS*. This follows from Pythagoras‘s theorem since *SX*^{2} = *PX*^{2} + *PS*^{2}.

The next construction which we examine is that to find a square equal in area to a given rectangle. We give the version as it appears in the Baudhayana Sulbasutra.

Consider the diagram on the right.

The rectangle *ABCD* is given. Let *L* be marked on *AD* so that *AL* = *AB*. Then complete the square *ABML*. Now bisect *LD* at *X* and divide the rectangle *LMCD* into two equal rectangles with the line *XY*. Now move the rectangle *XYCD* to the position *MBQN*. Complete the square *AQPX*.

Now the square we have just constructed is not the one we require and a little more work is needed to complete the work. Rotate *PQ* about *Q* so that it touches *BY* at *R*. Then *QP* = *QR* and we see that this is an ideal “rope” construction. Now draw *RE* parallel to *YP* and complete the square *QEFG*. This is the required square equal to the given rectangle *ABCD*.

The Baudhayana Sulbasutra offers no proof of this result (or any other for that matter) but we can see that it is true by using Pythagoras‘s theorem.

*EQ*

^{2}=

*QR*

^{2}–

*RE*

^{2}

=

*QP*

^{2}–

*YP*

^{2}

=

*ABYX*+

*BQNM*

=

*ABYX*+

*XYCD*

=

*ABCD*.

All the Sulbasutras contain a method to square the circle. It is an approximate method based on constructing a square of side ^{13}/_{15} times the diameter of the given circle as in the diagram on the right. This corresponds to taking π = 4 × (^{13}/_{15})^{2} = ^{676}/_{225} = 3.00444 so it is not a very good approximation and certainly not as good as was known earlier to the Babylonians.

It is worth noting that many different values of π appear in the Sulbasutras, even several different ones in the same text. This is not surprising for whenever an approximate construction is given some value of π is implied. The authors thought in terms of approximate constructions, not in terms of exact constructions with π but only having an approximate value for it. For example in the Baudhayana Sulbasutra, as well as the value of ^{676}/_{225}, there appears ^{900}/_{289} and ^{1156}/_{361}. In different Sulbasutras the values 2.99, 3.00, 3.004, 3.029, 3.047, 3.088, 3.1141, 3.16049 and 3.2022 can all be found; see [6]. In [3] the value π = ^{25}/_{8} = 3.125 is found in the Manava Sulbasutras.

In [9] in addition to examining the problem of squaring the circle as given by Apastamba, the authors examine the problem of dividing a segment into seven equal parts which occurs in the same Sulbasutra.

The Sulbasutras also examine the converse problem of finding a circle equal in area to a given square.

Consider the diagram on the right.

The following construction appears. Given a square *ABCD* find the centre *O*. Rotate *OD* to position *OE* where *OE* passes through the midpoint *P* of the side of the square *DC*. Let *Q* be the point on *PE* such that *PQ* is one third of *PE*. The required circle has centre *O* and radius *OQ*.

Again it is worth calculating what value of π this implies to get a feel for how accurate the construction is. Now if the square has side 2*a* then the radius of the circle is *r* where

*r*=

*OE*–

*EQ*

= √2

*a*–

^{2}/

_{3}(√2

*a*–

*a*)

=

*a*(

^{√2}/

_{3}+

^{2}/

_{3}).

Then 4*a* ^{2} = π*a*^{2} (^{√2}/_{3} + ^{2}/_{3})^{2}which gives π = 36/(√2 + 2)^{2} = 3.088.

As a final look at the mathematics of the Sulbasutras we examine what may be the most remarkable. Both the Apastamba Sulbasutra and the Katyayana Sulbasutra give the following approximation to √2:-

Increase a unit length by its third and this third by its own fourth less the thirty-fourth part of that fourth.

Now this gives

^{577}/

_{408}

which is, to nine places, 1.414215686. Compare the correct value √2 = 1.414213562 to see that the Apastamba Sulbasutra has the answer correct to five decimal places. Of course no indication is given as to how the authors of the Sulbasutras achieved this remarkable result. Datta, in 1932, made a beautiful suggestion as to how this approximation may have been reached.

In [1] Datta considers a diagram similar to the one on the right.

The most likely reason for the construction was to build an altar twice the size of one already built. Datta’s suggestion involves taking two squares and cutting up the second square and assembling it around the first square to give a square twice the size, thus having side √2. The second square is cut into three equal strips, and strips 1 and 2 placed around the first square as indicated in the diagram. The third strip has a square cut off the top and placed in position 3. We now have a new square but some of the second square remains and still has to be assembled around the first.

Cut the remaining parts (two-thirds of a strip) into eight equal strips and arrange them around the square we are constructing as in the diagram. We have now used all the parts of the second square but the new figure we have constructed is not quite a square having a small square corner missing. It is worth seeing what the side of this “not quite a square” is. It is

which, of course, is the first three terms of the approximation. Now Datta argues in [1] that to improve the “not quite a square” the Sulbasutra authors could have calculated how broad a strip one needs to cut off the left hand side and bottom to fill in the missing part which has area (^{1}/_{12})^{2}. If *x* is the width one cuts off then

*x*× (1 +

^{1}/

_{3}+

^{1}/

_{12}) = (

^{1}/

_{12})

^{2}.

This has the solution *x* = 1/(3 × 4 × 34) which is approximately 0.002450980392. We now have a square the length of whose sides is

which is exactly the approximation given by the Apastamba Sulbasutra.Of course we have still made an approximation since the two strips of breadth *x* which we cut off overlapped by a square of side *x* in the bottom left hand corner. If we had taken this into account we would have obtained the equation

*x*× (1 +

^{1}/

_{3}+

^{1}/

_{12}) –

*x*

^{2}= (

^{1}/

_{12})

^{2}

for *x* which leads to *x* = ^{17}/_{12} – √2 which is approximately equal to 0.002453105. Of course we cannot take this route since we have arrived back at a value for *x* which involves √2 which is the quantity we are trying to approximate!In [4] Gupta gives a simpler way of obtaining the approximation for √2 than that given by Datta in [1]. He uses linear interpolation to obtain the first two terms, he then corrects the two terms so obtaining the third term, then correcting the three terms obtaining the fourth term. Although the method given by Gupta is simpler (and an interesting contribution) there is certainly something appealing in Datta’s argument and somehow a feeling that this is in the spirit of the Sulbasutras.

Of course the method used by these mathematicians is very important to understanding the depth of mathematics being produced in India in the middle of the first millennium BC. If we follow the suggestion of some historians that the writers of the Sulbasutras were merely copying an approximation already known to the Babylonians then we might come to the conclusion that Indian mathematics of this period was far less advanced than if we follow Datta’s suggestion.

# Jaina mathematics

^{th}century BC. To a certain extent it began to replace the Vedic religions which, with their sacrificial procedures, had given rise to the mathematics of building altars. The mathematics of the Vedic religions is described in the article Indian Sulbasutras.Now we could use the term Jaina mathematics to describe mathematics done by those following Jainism and indeed this would then refer to a part of mathematics done on the Indian subcontinent from the founding of Jainism up to modern times. Indeed this is fair and some of the articles in the references refer to fairly modern mathematics. For example in [16] Jha looks at the contributions of Jainas from the 5

^{th}century BC up to the 18

^{th}century AD.

This article will concentrate on the period after the founding of Jainism up to around the time of Aryabhata in around 500 AD. The reason for taking this time interval is that until recently this was thought to be a time when there was little mathematical activity in India. Aryabhata‘s work was seen as the beginning of a new classical period for Indian mathematics and indeed this is fair. Yet Aryabhata did not work in mathematical isolation and as well as being seen as the person who brought in a new era of mathematical investigation in India, more recent research has shown that there is a case for seeing him also as representing the end-product of a mathematical period of which relatively little is known. This is the period we shall refer to as the period of Jaina mathematics.

There were mathematical texts from this period yet they have received little attention from historians until recent times. Texts, such as the *Surya Prajnapti* which is thought to be around the 4^{th}century BC and the *Jambudvipa Prajnapti* from around the same period, have recently received attention through the study of later commentaries. The *Bhagabati Sutra* dates from around 300 BC and contains interesting information on combinations. From about the second century BC is the *Sthananga Sutra* which is particularly interesting in that it lists the topics which made up the mathematics studied at the time. In fact this list of topics sets the scene for the areas of study for a long time to come in the Indian subcontinent. The topics are listed in [2] as:-

… the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

The ideas of the mathematical infinite in Jaina mathematics is very interesting indeed and they evolve largely due to the Jaina’s cosmological ideas. In Jaina cosmology time is thought of as eternal and without form. The world is infinite, it was never created and has always existed. Space pervades everything and is without form. All the objects of the universe exist in space which is divided into the space of the universe and the space of the non-universe. There is a central region of the universe in which all living beings, including men, animals, gods and devils, live. Above this central region is the upper world which is itself divided into two parts. Below the central region is the lower world which is divided into seven tiers. This led to the work described in [3] on a mathematical topic in the Jaina work, *Tiloyapannatti* by Yativrsabha. A circle is divided by parallel lines into regions of prescribed widths. The lengths of the boundary chords and the areas of the regions are given, based on stated rules.This cosmology has strongly influenced Jaina mathematics in many ways and has been a motivating factor in the development of mathematical ideas of the infinite which were not considered again until the time of Cantor. The Jaina cosmology contained a time period of 2^{588} years. Note that 2^{588} is a very large number!

^{588}= 1013 065324 433836 171511 818326 096474 890383 898005 918563 696288 002277 756507 034036 354527 929615 978746 851512 277392 062160 962106 733983 191180 520452 956027 069051 297354 415786 421338 721071 661056.

So what are the Jaina ideas of the infinite. There was a fascination with large numbers in Indian thought over a long period and this again almost required them to consider infinitely large measures. The first point worth making is that they had different infinite measures which they did not define in a rigorous mathematical fashion, but nevertheless are quite sophisticated. The paper [6] describes the way that the first unenumerable number was constructed using effectively a recursive construction.The Jaina construction begins with a cylindrical container of very large radius *r*^{q} (taken to be the radius of the earth) and having a fixed height *h*. The number *n*^{q} = *f*(*r*^{q}) is the number of very tiny white mustard seeds that can be placed in this container. Next, *r*_{1} = *g*(*r*^{q}) is defined by a complicated recursive subprocedure, and then as before a new larger number *n*_{1} = *f*(*r*_{1}) is defined. The text the *Anuyoga Dwara Sutra* then states:-

Still the highest enumerable number has not been attained.

The whole procedure is repeated, yielding a truly huge number which is called jaghanya- parita- asamkhyata meaning “unenumerable of low enhanced order”. Continuing the process yields the smallest unenumerable number.Jaina mathematics recognised five different types of infinity [2]:-

… infinite in one direction, infinite in two directions, infinite in area, infinite everywhere and perpetually infinite.

The *Anuyoga Dwara Sutra* contains other remarkable numerical speculations by the Jainas. For example several times in the work the number of human beings that ever existed is given as 2^{96}.By the second century AD the Jaina had produced a theory of sets. In *Satkhandagama* various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets.

Permutations and combinations are used in the *Sthananga Sutra.* In the *Bhagabati Sutra* rules are given for the number of permutations of 1 selected from n, 2 from n, and 3 from n. Similarly rules are given for the number of combinations of 1 from n, 2 from n, and 3 from n. Numbers are calculated in the cases where n = 2, 3 and 4. The author then says that one can compute the numbers in the same way for larger n. He writes:-

In this way,5,6,7, …,10, etc. or an enumerable, unenumerable or infinite number of may be specified. Taking one at a time, two at a time, … ten at a time, as the number of combinations are formed they must all be worked out.

Interestingly here too there is the suggestion that the arithmetic can be extended to various infinite numbers. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion was noted. In a commentary on this third century work in the tenth century, Pascal‘s triangle appears in order to give the coefficients of the binomial expansion.Another concept which the Jainas seem to have gone at least some way towards understanding was that of the logarithm. They had begun to understand the laws of indices. For example the *Anuyoga Dwara Sutra* states:-

The first square root multiplied by the second square root is the cube of the second square root.

The second square root was the fourth root of a number. This therefore is the formula

*a*).(√√

*a*) = (√√

*a*)

^{3}.

Again the *Anuyoga Dwara Sutra* states:-

… the second square root multiplied by the third square root is the cube of the third square root.

The third square root was the eighth root of a number. This therefore is the formula

*a*).(√√√

*a*) = (√√√

*a*)

^{3}.

Some historians studying these works believe that they see evidence for the Jainas having developed logarithms to base 2.The value of π in Jaina mathematics has been a topic of a number of research papers, see for example [4], [5], [7], and [17]. As with much research into Indian mathematics there is interest in whether the Indians took their ideas from the Greeks. The approximation π = √10 seems one which was frequently used by the Jainas.

Finally let us comment on the Jaina’s astronomy. This was not very advanced. It was not until the works of Aryabhata that the Greek ideas of epicycles entered Indian astronomy. Before the Jaina period the ideas of eclipses were based on a demon called Rahu which devoured or captured the Moon or the Sun causing their eclipse. The Jaina school assumed the existence of two demons Rahu, the Dhruva Rahu which causes the phases of the Moon and the Parva Rahu which has irregular celestial motion in all directions and causes an eclipse by covering the Moon or Sun or their light. The author of [23] points out that, according to the Jaina school, the greatest possible number of eclipses in a year is four.

Despite this some of the astronomical measurements were fairly good. The data in the *Surya Prajnapti* implies a synodic lunar month equal to 29 plus ^{16}/_{31} days; the correct value being nearly 29.5305888. There has been considerable interest in examining the data presented in these Jaina texts to see if the data originated from other sources. For example in the *Surya Prajnapti* data exists which implies a ratio of 3:2 for the maximum to the minimum length of daylight. Now this is not true for India but is true for Babylonia which makes some historians believe that the data in the *Surya Prajnapti* is not of Indian origin but is Babylonian. However, in [22] Sharma and Lishk present an alternative hypothesis which would allow the data to be of Indian origin. One has to say that their suggestion that 3:2 might be the ratio of the amounts of water to be poured into the water-clock on the longest and shortest days seems less than totally convincing.

## The Bakhshali manuscript

The Bakhshali Manuscript is the name given to the mathematical work written on birch bark and found in the summer of1881near the village Bakhshali(or Bakhshalai)of the Yusufzai subdivision of the Peshawar district(now in Pakistan). The village is in Mardan tahsil and is situated50miles from the city of Peshawar.An Inspector of Police named Mian An-Wan-Udin(whose tenant actually discovered the manuscript while digging a stone enclosure in a ruined place)took the work to the Assistant Commissioner at Mardan who intended to forward the manuscript to Lahore Museum. However, it was subsequently sent to the Lieutenant Governor of Punjab who, on the advice of General A Cunningham, directed it to be passed on to Dr Rudolf Hoernle of the Calcutta Madrasa for study and publication. Dr Hoernle presented a description of the BM before the Asiatic Society of Bengal in1882, and this was published in the Indian Antiquary in1883. He gave a fuller account at the Seventh Oriental Conference held at Vienna in1886and this was published in its Proceedings. A revised version of this paper appeared in the Indian Antiquary of1888. In1902, he presented the Bakhshali Manuscript to the Bodleian Library, Oxford, where it is still(Shelf mark: MS. Sansk. d.14).

A large part of the manuscript had been destroyed and only about 70 leaves of birch-bark, of which a few were only scraps, survived to the time of its discovery.To show the arguments regarding its age we note that F R Hoernle, referred to in the quotation above, placed the Bakhshali manuscript between the third and fourth centuries AD. Many other historians of mathematics such as Moritz Cantor, F Cajori, B Datta, S N Sen, A K Bag, and R C Gupta agreed with this dating. In 1927-1933 the Bakhshali manuscript was edited by G R Kaye and published with a comprehensive introduction, an English translation, and a transliteration together with facsimiles of the text. Kaye claimed that the manuscript dated from the twelfth century AD and he even doubted that it was of Indian origin.

Channabasappa in [6] gives the range 200 – 400 AD as the most probable date. In [5] the same author identifies five specific mathematical terms which do not occur in the works of Aryabhata and he argues that this strongly supports a date for the Bakhshali manuscript earlier than the 5^{th} century. Joseph in [3] suggests that the evidence all points to the:-

… manuscript[being]probably a later copy of a document composed at some time in the first few centuries of the Christian era.

L V Gurjar in [1] claims that the manuscript is no later than 300 AD. On the other hand T Hayashi claims in [2] that the date of the original is probably from the seventh century, but he also claims that the manuscript itself is a later copy which was made between the eighth and the twelfth centuries AD.I [EFR] feel that if one weighs all the evidence of these experts the most likely conclusion is that the manuscript is a later copy of a work first composed around 400 AD. Why do I believe that the actual manuscript was written later? Well our current understanding of Indian numerals and writing would date the numerals used in the manuscript as not having appeared before the ninth or tenth century. To accept that this style of numeral existed in 400 AD. would force us to change greatly our whole concept of the time-scale for the development of Indian numerals. Sometimes, of course, we are forced into major rethinks but, without supporting evidence, everything points to the manuscript being a tenth century copy of an original from around 400 AD. Despite the claims of Kaye, it is essentially certain that the manuscript is Indian.

The attraction of the date of 400 AD for the Bakhshali manuscript is that this puts it just before the “classical period” of Indian mathematics which began with the work of Aryabhata I around 500. It would then fill in knowledge we have of Indian mathematics for, prior to the discovery of this manuscript, we had little knowledge of Indian mathematics between the dates of about 200 BC. and 500 AD. This date would make it a document near the end of the period of Jaina mathematics and it can be seen as, in some sense, marking the achievements of the Jains.

What does the manuscript contain? Joseph writes in [3]:-

The Bakhshali manuscript is a handbook of rules and illustrative examples together with their solutions. It is devoted mainly to arithmetic and algebra, with just a few problems on geometry and mensuration. Only parts of it have been restored, so we cannot be certain about the balance between different topics.

Now the way that the manuscript is laid out is quite unusual for an Indian document (which of course leads people like Kaye to prefer the hypothesis that it is not Indian at all – an idea in which we cannot see any merit). The Bakhshali manuscript gives the statement of a rule. There then follows an example given first in words, then using mathematical notation. The solution to the example is then given and finally a proof is set out.The notation used is not unlike that used by Aryabhata but it does have features not found in any other document. Fractions are not dissimilar in notation to that used today, written with one number below the other. No line appears between the numbers as we would write today, however. Another unusual feature is the sign + placed after a number to indicate a negative. It is very strange for us today to see our addition symbol being used for subtraction. As an example, here is how ^{3}/_{4} – ^{1}/_{2} would be written.

^{3}/_{4} minus ^{1}/_{2}

Compound fractions were written in three lines. Hence 1 plus ^{1}/_{3} would be written thus

**1 plus ^{1}/_{3}**

and 1 minus ^{1}/_{3} = ^{2}/_{3} in the following way

**1 minus ^{1}/_{3} = ^{2}/_{3}**

Sums of fractions such as ^{5}/_{1} plus ^{2}/_{1} are written using the symbol yu ( for yuta)

^{5}/_{1} plus ^{2}/_{1}

Division is denoted by bha, an abbreviation for bhaga meaning “part”. For example

**8 divided by ^{2}/_{3}**

Equations are given with a large dot representing the unknown. A confusing aspect of Indian mathematics is that this notation was also often used to denote zero, and sometimes this same notation for both zero and the unknown are used in the same document. Here is an example of an equation as it appears in the Bakhshali manuscript.

**Equation**

The method of equalisation is found in many types of problems which occur in the manuscript. Problems of this type which are found in the manuscript are examined in [9] and some of these lead to indeterminate equations. Included are problems concerning equalising wealth, the positions of two travellers, wages, and purchases by a number of merchants. These problems can all be reduced to solving a linear equation with one unknown or to a system of n linear equations in n unknowns. To illustrate we give the following indeterminate problem which, of course, does not have a unique solution:-

One person possesses seven asava horses, another nine haya horses, and another ten camels. Each gives two animals, one to each of the others. They are then equally well off. Find the price of each animal and the total value of the animals possesses by each person.

The solution, translated into modern notation, proceeds as follows. We seek integer solutions *x*_{1}, *x*_{2}, *x*_{3} and *k* (where *x*_{1} is the price of an asava, *x*_{2} is the price of a haya, and *x*_{3} is the price of a horse) satisfying

*x*

_{1}+

*x*

_{2}+

*x*

_{3}=

*x*

_{1}+ 7

*x*

_{2}+

*x*

_{3}=

*x*

_{1}+

*x*

_{2}+ 8

*x*

_{3}=

*k*.

Then 4 *x*_{1} = 6 *x*_{2} = 7 *x*_{3} = *k* – (*x*_{1} + *x*_{2} + *x*_{3}).For integer solutions *k* – (*x*_{1} + *x*_{2} + *x*_{3}) must be a multiple of the lcm of 4, 6 and 7. This is the indeterminate nature of the problem and taking different multiples of the lcm will lead to different solutions. The Bakhshali manuscript takes *k* – (*x*_{1} + *x*_{2} + *x*_{3}) = 168 (this is 4 × 6 × 7) giving *x*_{1} = 42, *x*_{2} = 28, *x*_{3} = 24. Then *k* = 262 is the total value of the animals possesses by each person. This is not the minimum integer solution which would be *k* = 131.

If we use modern methods we would solve the system of three equation for *x*_{1}, *x*_{2}, *x*_{3} in terms of *k* to obtain

*x*

_{1}= 21

*k*/131,

*x*

_{2}= 14

*k*/131,

*x*

_{3}= 12

*k*/131

so we obtain integer solutions by taking *k* = 131 which is the smallest solution. This solution is not given in the Bakhshali manuscript but the author of the manuscript would have obtained this had he taken *k* – (*x*_{1} + *x*_{2} + *x*_{3}) = lcm(4, 6, 7) = 84.Here is another equalisation problem taken from the manuscript which has a unique solution:-

Two page-boys are attendants of a king. For their services one gets13/6dinaras a day and the other3/2. The first owes the second10dinaras. calculate and tell me when they have equal amounts.

Now I would solve this by saying that the first gets ^{13}/_{6} – ^{3}/_{2} = ^{2}/_{3} dinaras more than the second each day. He needs 20 dinaras more than the second to be able to give back his 10 dinaras debt and have them with equal amounts. So 30 days are required when each has 13 × ^{30}/_{6} – 10 = 55 dinaras. This is not the method of the Bakhshali manuscript which uses the “rule of three”.The rule of three is the familiar way of solving problems of the type: if a man earns 50 dinaras in 8 days how much will he earn in 12 days. The Bakhshali manuscript describes the rule where the three numbers are written

The 8 is the “pramana”, the 50 is the “phala” and the 12 is the “iccha”. The rule, according to the Bakhshali manuscript gives the answer as

or in the case of the example 50 × ^{12}/_{8} = 75 dinaras.Applying this to the page-boy problem we obtain equal amounts for the page-boys after *n* days where

*n*/

_{6}= 3 ×

*n*/

_{2}+20

so *n* = 30 and each has 13 × ^{30}/_{6} – 10 = 55 dinaras.Another interesting piece of mathematics in the manuscript concerns calculating square roots. The following formula is used

*Q*= √(

*A*

^{2}+

*b*) =

*A*+

*b*/2

*A*– (

*b*/2

*A*)

^{2}/(2(

*A*+

*b*/2

*A*))

This is stated in the manuscript as follows:-

In the case of a non-square number, subtract the nearest square number, divide the remainder by twice this nearest square; half the square of this is divided by the sum of the approximate root and the fraction. this is subtracted and will give the corrected root.

Taking *Q* = 41, then *A* = 6, *b* = 5 and we obtain 6.403138528 as the approximation to √41 = 6.403124237. Hence we see that the Bakhshali formula gives the result correct to four decimal places.The Bakhshali manuscript also uses the formula to compute √105 giving 10.24695122 as the approximation to √105 = 10.24695077. This time the Bakhshali formula gives the result correct to five decimal places.

The following examples also occur in the Bakhshali manuscript where the author applies the formula to obtain approximate square roots:

Bakhshali formula gives 22.068076490965

Correct answer is 22.068076490713

Here 9 decimal places are correct√889

Bakhshali formula gives 29.816105242176

Correct answer is 29.8161030317511

Here 5 decimal places are correct

[Note. If we took 889 = 30^{2} – 11 instead of 29^{2} + 48 we would get

Bakhshali formula gives 29.816103037078

Correct answer is 29.8161030317511

Here 8 decimal places are correct]

√339009

Bakhshali formula gives 582.2447938796899

Correct answer is 582.2447938796876

Here 11 decimal places are correct

It is interesting to note that Channabasappa [6] derives from the Bakhshali square root formula an iterative scheme for approximating square roots. He finds in [7] that it is 38% faster than Newton‘s method in giving √41 to ten places of decimals.

## Indian Mathematics: Redressing the balance

## 1: Abstract

Mathematics has long been considered an invention of European scholars, as a result of which the contributions of non-European countries have been severely neglected in histories of mathematics. Worse still, many key mathematical developments have been wrongly attributed to scholars of European origin. This has led to so-called Eurocentrism. The neglect of non-European mathematics is no more apparent than when studying the contributions of India. Contrary to Euroscentric belief, scholars from India, over a period of some 4500 years, contributed to some of the greatest mathematical achievements in the history of the subject. From the earliest numerate civilisation of the Indus valley, through the scholars of the 5^{th} to 12^{th} centuries who were conversant in arithmetic, algebra, trigonometry, geometry combinatorics and latterly differential calculus, Indian scholars led the world in the field of mathematics. The peak coming between the 14^{th} and 16^{th}centuries in the far South, where scholars were the first to derive infinite series expansions of trigonometric functions.

In addition to mighty contributions to all the principal areas of mathematics, Indian scholars were responsible for the creation, and refinement of the current decimal place value system of numeration, including the number zero, without which higher mathematics would not be possible. The purpose of my project is to highlight the major mathematical contributions of Indian scholars and further to emphasise where neglect has occurred and hence elucidate why the Eurocentric ideal is an injustice and in some cases complete fabrication.

## 2: Introduction

The history of science, and specifically mathematics, is a vast topic and one which can never be completely studied as much of the work of ancient times remains undiscovered or has been lost through time. Nevertheless there is much that is known and many important discoveries have been made, especially over the last 150 years, which have significantly altered the chronology of the history of mathematics, and the conceptions that had been commonly held prior to that. By the turn of the 21^{st} century it was fair to say that there was definite knowledge of where and when a vast majority of the significant developments of mathematics occurred. However, despite widely available, reliable information, I became aware during the course of more general studies of the history of mathematics, that many discrepancies persist.

I became drawn to the topic of Indian mathematics, as there appeared to be a distinct and inequitable neglect of the contributions of the sub-continent. Thus, during the course of this project I aim to discuss that despite slowly changing attitudes there is still an ideology’ which plagues much of the recorded history of the subject. That is, to some extent very little has changed even in our seemingly enlightened historical and cultural position, and, in specific reference to my study area, many of the developments of Indian mathematics remain almost completely ignored, or worse, attributed to scholars of other nationalities, often European.

It is important for me to clarify at this point that the ideology I refer to is that held by (predominantly) European historians of science and mathematics, that mathematics is a European ‘invention’. This ideology leads to an ‘intrinsically’ Eurocentric bias to the history of the subject. As R Rashed comments the ideology can be summarised as such:

…Classical science is European and its origins are directly traceable to Greek philosophy and science.[RR, P 332]

Thus despite the many discoveries that have been made there has been a great reluctance to acknowledge the contributions of non-European countries. For several hundred years following the European mathematical renaissance of the late 15^{th} and early 16^{th} century there was a commonly held opinion among commentators and historians that mathematics originated in its entirety from Europe and European scholars.

The basic chronology of the history of mathematics was very simple; it had primarily been the invention of the ancient Greeks, whose work had continued up to the middle of the first millenium A.D. Following which there was a period of almost 1000 years where no work of significance was carried out until the European renaissance, which coincided with the ‘reawakening’ of learning and culture in Europe following the so called dark ages.

Figure 2.1: Eurocentric chronology of mathematics history.

Some historians made some concessions, by acknowledging the work of Egyptian, Babylonian, Indian and Arabic mathematicians (and occasionally the work of the Far East and China). Modified versions of the Eurocentric model commonly took the form seen below.

Figure 2.2: Modified Eurocentric model.

However, references to the work of these ‘others’ or ‘non-Europeans’ were always brief and hazy, and generally concluded that they were merely reconstructions of Greek works and that nothing of significance or importance was contained in them. Indian scholars, on the relatively rare occasions they were discussed, were merely considered to be custodians of Ancient Greek learning. There is a plethora of works and quotes that highlight the attitude that prevailed towards the works of so called non-European mathematics. P Duhem (“Le systeme du monde”, 1965) states very simply:

…Arabic Science only reproduced the teachings received from Greek science.[RR, P 338]

Furthermore G Sarton (“*Guide to the History of Science*“, 1927) comments:

…One could almost omit Hindu and Chinese developments in Mathematics.[AA’D, P 15]

While P Tannery (“*La geometrie grecque*“, 1887) opines:

…The more one examines the Hindu scholars the more they appear dependent upon the Greeks…(and)…quite inferior to their predecessors on all respects.[RR, P 338]

It is this underhand and in some cases completely fallacious attitude towards Indian mathematics that I have chosen to focus upon.

As said this is a vast topic area and although all non-European ‘roots’ of mathematics have suffered neglect and miss-representation by many historians I am not going to attempt to focus on all non-European roots of mathematical development. I have chosen to focus on the mathematical developments of the Indian sub-continent, as I consider them not only to be severely neglected in histories of mathematics, but also to have produced some of the most remarkable results of mathematics. Indeed, the research I have conducted has highlighted that many Indian mathematical results, beyond being simply remarkable because of the time in which they were derived, show that several ‘key’ mathematical topics, and subsequent results, indubitably originate from the Indian subcontinent.

Having made these discoveries it seems to me an incredible injustice that the work of Indian scholars is not rewarded with a much more prominent place in the history of mathematics.

Not only have many historians identified Greek influences in all of the work of Indian scholars, (in some cases by suggesting severely inaccurate dates for the works), several others have even attempted to show Arabic influences, which is quite incredible as several Indian works in fact had a significant influence on early Arab works. Although I by no means wish to talk down the incredible developments of Arab scholars, I believe the developments of Indian scholars are on a par with, and occasionally surpass them.

Indian scholars made vast contributions to the field of mathematical astronomy and as a result contributed mightily to the developments of *arithmetic*, *algebra*, *trigonometry* and secondarily *geometry* (although this topic was well developed by the Greeks) and *combinatorics*. Perhaps most remarkable were developments in the fields of *infinite series expansions of trigonometric expressions* and *differential calculus*.

Surpassing all these achievements however was the development of *decimal numeration* and the *place value system*, which without doubt stand together as the most remarkable developments in the history of mathematics, and possibly one of the foremost developments in the history of humankind. The decimal place value system allowed the subject of mathematics to be developed in ways that simply would not have been possible otherwise. It also allowed numbers to be used more extensively and by vastly more people than ever before.

The aim of my work is not just to paint as accurate a picture of the developments of mathematics by the Indian peoples as possible, but also to attempt to give reasons as to why Europe has chosen to neglect the facts of history. Through a detailed discussion of the mathematics of the Indian subcontinent I hope to highlight why this ‘neglect’ is such an injustice, and briefly discuss the possible consequences of this neglect.

There are several points that I feel it is important to make before progressing further with my discussion. The first is to make it clear that the chronology of the history of mathematics is not entirely linear. One will often find simplified diagrammatic representations where the work of one group of people (or country) is proceeded by the work of another group and so on. In reality things are far more complicated than this. Particularly in the European dark ages (5^{th}-15^{th} centuries) mathematical developments passed between several countries, being constantly refined and improved. G Joseph states:

… A variety of mathematical activity and exchange between a number of cultural areas went on while Europe was in a deep slumber.[GJ, P 9]

Figure 2.3: Non-European mathematics during the dark ages.

Secondly it is worth noting that we view the history of mathematics from:

…Our own position of understanding and sophistication.[EFR/JO’C1, P 3]

While this is unavoidable it is vital to appreciate the uniqueness and ingenuity of the developments of Indian scholars, even if the results are common-place now.

We also view mathematics from an intrinsically European standpoint due largely to the influence of European scholars over the last 500 years and the colonisation of much of the world by European countries. Perhaps this is why many historians find it hard to accept that many results and developments of mathematics are not European in origin. In short, if we are European, somewhat unavoidably we view history from our indigenous standpoint.

There is also a slight methodological problem related to the ‘labeling’ of the topic of “Indian mathematics” that I will briefly discuss. Using the label “Indian mathematics” is not entirely accurate as much of the earliest “Indian mathematics” was developed in areas, which are now part of Pakistan. The label is used for simplicity and can be justified by stating that developments took place in the Indian subcontinent. Quite often the ‘label’ Hindu mathematics is used, but it is less accurate as many of the scholars were not Hindus.

Finally before progressing I will specify the time period that my work will cover. The earliest origins of Indian mathematics have been dated to around 3000 BC and this seems a sensible point at which to commence my discussion, while work of a significant nature was still being carried out in the south of India in the 16^{th} century, following which there was an eventual decline. It is hence a vast time scale of almost 5000 years, and indeed it may be greater than that, the estimation of 3000 BC is a slightly crude approximation, and there remains much controversy with regards to the dating of many works prior to 400 AD.

As Gupta states in his paper on the problem of ancient Indian chronology:

…In the case of India, the problem of chronology continues to be very serious especially with regard to the prehistoric and ancient periods.[RG1, P 17]

It is also worth pointing out that this lack of certainty has allowed several unscrupulous scholars to pick dates of choice for certain Indian discoveries so as to justify suggestions for Greek, Arab or other influences. In situations where this has arisen I will attempt to the best of my ability to state fact, although in some cases a well-informed guess will have to suffice.

I will now commence with the main body of my work, a discussion of the development of mathematics, as a subject, in India, through which I hope to highlight (as previously stated) both the many remarkable discoveries, and results, and where neglect or incorrect analysis has occurred.

Figure 2.4: Map of India. [GJ, P 220]

## 3: Early Indian culture – Indus civilisation

The first appearance of evidence of the use of mathematics in the Indian subcontinent was in the Indus valley (see Figure 2.4) and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilisation was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced ‘brick technology’, (which utilised ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding.

The discoveries of systems of uniform and decimal weights, over a vast area, are of considerable interest. G Joseph states:

…Such standardisation and durability is a strong indication of a numerate culture.[GJ, P 222]

Also, many of the weights uncovered have been produced in definite geometrical shapes (cuboid, barrel, cone, and cylinder to name a few) which present knowledge of basic geometry, including the circle.

This culture also produced artistic designs of a mathematical nature and there is evidence on carvings that these people could draw concentric and intersecting circles and triangles, leading S Sinha to state:

…The civilisation and culture of the inhabitants of the Indus valley…were of a very advanced nature.[SS1, P 71]

S Srinivasan further comments:

…There are many unique features in the construction patterns, which suggest an independent origin of ideas in ancient Indian civilisation.[SSr1, P17]

Further to the use of circles in ‘decorative’ design there is indication of the use of bullock carts, the wheels of which may have had a metallic band wrapped round the rim. This clearly points to the possession of knowledge of the ratio of the length of the circumference of the circle and its diameter, and thus values of p.

Also of great interest is a remarkably accurate decimal ruler known as the Mohenjodaro ruler. Subdivisions on the ruler have a maximum error of just 0.005 inches and, at a length of 1.32 inches, have been named the Indus inch. Furthermore, a correspondence has been noted between the Indus scale and brick size. Bricks (found in various locations) were found to have dimensions that were integral multiples of the graduations of their respective scales, which suggests advanced mathematical thinking.

Figure 3.1: Ruler found at Lothal. [SSr1, P17]

Above all else there are also brief references to an early decimal system of numeration. The seeds of what were to become the single greatest contribution of the Indian sub-continent to the world (not just of mathematics) had already been sown. My evidence comes from S Sinha who states:

…Writers on these civilisations briefly refer to the decimal system of numeration found in these excavations.[SS1, P 71]

This quote supports the theory that the Brahmi numerals, which were to go on to develop into the numerals we use today, originated in the Indus valley around 2000 BC, however this theory has been rejected by several scholars including Ifrah and Joseph. This quote could be considered a piece of overzealous reporting by the author however, on further investigation I can support the comment with some confidence.

Not only are the markings on all the excavated measuring devices decimal in nature, but there is also research currently being conducted, which is attempting, with success, to show a connection between the Brahmi and Indus scripts. This lends indirect support to suggestions of the existence of early decimal numeral forms. As I will discuss briefly later, the Brahmi numerals undoubtedly developed into the numeral forms we use today.

Although this early mathematics is generally included in histories of mathematics it is often in nothing more than a brief mention, and there is a most curious quote by J Katz who claims:

…There is no direct evidence of its(Harappan civilisation)mathematics.[JK, P4]

It is possible that he makes this comment with regards to the fact that the Indus script as yet remains undeciphered (GJ, P218).

However R Gupta more ‘sensibly’ states:

…In fact the level of mathematical knowledge implied in various geometrical designs, accurate layout of streets and drains and various building constructions etc was quite high (from a practical point of view).[RG1, P131]

While Childe claims:

…India confronts Egypt and Babylonia by the 3[EFR/JJO’C2, P 1]^{rd}millennium with a thoroughly individual and independent civilisation of her own.

Some confusion exists as to what caused the decline of this Harrapan culture, there are several theories, the most probable of which in my opinion was the drying up of the Sarasvati River. This view is supported by S Kak and also S Kalyanaraman who has written an extensive paper on the topic and comments:

.

.. The drying-up of the Sarasvati River led to migrations of people eastwards.

The most commonly held view by historians is that Aryan peoples from the North invaded and destroyed the Harappan culture, this view however is considered increasingly contentious. In addition to the significance the fledgling decimal system would ultimately have, the most important legacy of this early civilisation is the influence its brick technology *may* have had on the altar building required by the Vedic religion that followed. A theory of the ‘interlinkage’ of the Harappan and Vedic cultures has recently arisen from a variety of studies, and it may come to light that there was a greater interaction between the two civilisations than currently thought.

## 4: Mathematics in the service of religion: I. Vedas and Vedangas

The Vedic religion was followed by the Indo-Aryan peoples, who originated from the north of the sub-continent. It is through the works of Vedic religion that we gain the first literary evidence of Indian culture and hence mathematics. Written in Vedic Sanskrit the Vedic works, Vedas and Vedangas (and later Sulbasutras) are primarily religious in content, but embody a large amount of astronomical knowledge and hence a significant knowledge of mathematics. The requirement for mathematics was (at least at first) twofold, as R Gupta discusses:

…The need to determine the correct times for Vedic ceremonies and the accurate construction of altars led to the development of astronomy and geometry.[RG2, P 131]

Some chronological confusion exists with regards to the appearance of the Vedic religion. S Kak states in a very recent work that the time period for the Vedic religion stretches back potentially as far as 8000BC and definitely 4000BC. Whereas G Joseph states 1500 BC as the forming of the Hindu civilisation and the recording of Vedas and Vedangas, and later Sulbasutras. However it seems most likely that significant knowledge of astronomy and mathematics first appears in Vedic works around the 2^{nd} millennium BC. The *Rg-Veda* (fire altar) the earliest extant Vedic work dates from around 1900 BC. R Gupta in his paper on the problem of ancient Indian chronology shows that dates from 26000-200 BC have been suggested for the Vedic ‘period’. Having consulted many sources I am confident at placing the period of the Vedas (and Vedangas) at around 1900-1000 BC.

Further mathematical work is found in the Sulbasutras of the later Vedic period, the earliest of which is thought to have been written around 800 BC and the last around 200 BC. I will now move on from this slightly clouded chronological discussion. It is however worth noting that there are serious underlying problems with the chronology of early Indian mathematics which require significant attention.

Although the requirement of mathematics at this time was clearly not for its own sake, but for the purposes of religion and astronomy, it is important not to ignore the secular use of the texts, i.e. by the craftsmen who were building the altars. Similarly with the earlier Harappan peoples it seems likely that (at least) basic mathematics will have grown to become used by large numbers of the population. Regardless of the fact that at this time mathematics remained for practical uses, some significant work in the fields of geometry and arithmetic were developed during the Vedic period and as L Gurjar states:

…The Hindu had made enormous strides in the field of mathematics.[LG, P 2]

It is also worthwhile briefly noting the astronomy of the Vedic period which, given very basic measuring devices (in many cases just the naked eye), gave surprisingly accurate values for various astronomical quantities. These include the relative size of the planets the distance of the earth from the sun, the length of the day, and the length of the year. For further information, S Kak is an authority on the astronomical content of Vedic works.

Much of the mathematics contained within the Vedas is found in works called Vedangas of which there are six. Of the six Vedangas those of particular significance are the Vedangas *Jyotis* and *Kalpa* (the fifth and sixth Vedangas). Jyotis was (at the time) the name for astronomy, while Kalpa contained the rules for the rituals and ceremonies. The Vedangas are best described as an auxiliary to the Vedas.

N Dwary claims, with reference to the Vedanga-Jyotis, that:

…Hindus of the period were fully conversant with fundamental operations of arithmetic.[ND1, P 39]

S Kak suggests a date of around 1350 BC for the Vedanga-Jyotis. I include this as a reminder of the time period being discussed.

Along with the Vedangas there are several further works that contain mathematics, including:

Taittiriya SamhitaSatapatha BrahmanaandYajur and Atharva-VedaRg-Veda(of which it is thought there are three ‘versions’) plus additionalSamhitas

Of these the Taittiriya Samhita and Rg-Veda are considered the oldest and contain rules for the construction of great fire altars.

Figure 4.1.1: First layer of a Vedic sacrificial altar (in the shape of a falcon). [GJ, P 227]

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

Use of geometric shapes, including triangles, rectangles, squares, trapezia and circles.

Equivalence through numbers and area.

Equivalence led to the problem of:

Squaring the circle and visa-versa.

Early forms of Pythagoras theorem.

Estimations for p.

S Kak gives three values for p from the *Satapatha Brahmana*. It seems most probable that they arose from transformations of squares into circles and circles to squares. The values are:

p_{1} = ^{25}/_{8} (3.125)

p_{2} = ^{900}/_{289} (3.11418685…)

p_{3} = ^{1156}/_{361} (3.202216…)

Astronomical calculations also leads to a further Vedic approximation:

p_{4} = ^{339}/_{108} (3.1389)

This is correct (when rounded) to 2 decimal places.

Also found in Vedic works are:

All four arithmetical operators (addition, subtraction, multiplication and division).

A definite system for denoting any number up to 1055 and existence of zero.

Prime numbers.

The Arab scholar Al-Biruni (973-1084 AD) discovered that only the Indians had a number system that was capable of going beyond the thousands in naming the orders in decimal counting.

Evidence of the use of this advanced numerical concept leads S Sinha to comment:

…It is fair to agree that a nation with such an advanced and cultured civilisation and which was using the numerical system (decimal place value) knew also how to handle the associated arithmetic.[SS1, P 73]

It is in Vedic works that we also first find the term “*ganita*” which literally means “the science of calculation”. It is basically the Indian equivalent of the word mathematics and the term occurs throughout Vedic texts and in all later Indian literature with mathematical content.

Among the other works I have mentioned, mathematical material of considerable interest is found:

Arithmetical sequences, the decreasing sequence 99, 88, … , 11 is found in the

Atharva-Veda.

Pythagoras’s theorem, geometric, constructional, algebraic and computational aspects known. A rule found in theSatapatha Brahmanagives a rule, which implies knowledge of the Pythagorean theorem, and similar implications are found in the Taittiriya Samhita.

Fractions, found in one (or more) of theSamhitas.

Equations, 972x^{2}= 972 +mfor example, found in one of theSamhitas.

The ‘rule of three’.

## 4 II. Sulba Sutras

The later Sulba-sutras represent the ‘traditional’ material along with further related elaboration of Vedic mathematics. The Sulba-sutras have been dated from around 800-200 BC, and further to the expansion of topics in the Vedangas, contain a number of significant developments.

These include first ‘use’ of irrational numbers, quadratic equations of the form *a* *x*^{2} = *c* and *ax*^{2} + *bx* = *c*, unarguable evidence of the use of Pythagoras theorem and Pythagorean triples, *predating*Pythagoras (c 572 – 497 BC), and evidence of a number of geometrical proofs. This is of great interest as proof is a concept thought to be completely lacking in Indian mathematics.

Example 4.2.1: Pythagoras theorem and Pythagorean triples, as found in the Sulba Sutras.

The rope stretched along the length of the diagonal of a rectangle makes an area which the, vertical and horizontal sides make together.

In other words:

a^{2}=b^{2}+c^{2}Examples of Pythagorean triples given as the sides of right angled triangles:

5, 12, 13

8, 15, 17

12, 16, 20

12, 35, 37

Of the Sulvas so far ‘uncovered’ the four major and most mathematically significant are those composed by Baudhayana, Manava, Apastamba and Katyayana (perhaps least ‘important’ of the Sutras, by the time it was composed the Vedic religion was becoming less predominant). However in a paper written 20 years ago S Sinha claims that there are a further three Sutras, ‘composed’ by Maitrayana, Varaha and Vadhula (SS1, P 76). I have yet to come across any other references to these three ‘extra’ sutras. These men were not mathematicians in the modern sense but they are significant none the less in that they were the first mentioned ‘individual’ composers. E Robertson and J O’Connor have suggested that they were Vedic priests (and skilled craftsmen).

It is thought that the Sulvas were intended to supplement the *Kalpa* (the sixth Vedanga), and their primary content remained instructions for the construction of sacrificial altars. The name Sulvasutra means ‘rule of chords’ which is another name for geometry.

N Dwary states:

…They offer a wealth of geometrical as well as arithmetical results.[ND1, P 40]

R Gupta similarly claims:

…The Sulba-sutras are (quite) rich in mathematical contents.[RG2, P 133]

With reference to the possible appearance of proof is a quote from A Michaels:

.

..Vedic geometry, though non-axiomatic in character, is provable and indeed[RG2, P 133]proof is implicitin several constructions prescribed in the Sulba-sutras.

This is not particularly compelling evidence but does suggest that the composers of the sulba-sutras may have had a greater depth of knowledge than is generally thought.

Many suggestions for the value of p are found within the sutras. They cover a surprisingly wide range of values, from 2.99 to 3.2022.

Pythagoras’s theorem and Pythagorean triples arose as the result of geometric rules. It is first found in the Baudhayana sutra – so was hence known from around 800 BC. It is also implied in the later work of Apastamba, and Pythagorean triples are found in his rules for altar construction. Altar construction also led to the discovery of irrational numbers, a remarkable estimation of 2 in found in three of the sutras. The method for approximating the value of 2 gives the following result:

2 = 1 +

^{1}/_{3}+^{1}/_{3.4}–^{1}/_{3.4.34}

This is equal to 1.412156…, which is correct to 5 decimal places.

It has been argued by scholars seemingly attempting to deprive Indian mathematics of due credit, that Indians believed that 2 = 1 + ^{1}/_{3} + ^{1}/_{3.4} – ^{1}/_{3.4.34}*exactly*, which would not indicate knowledge of the concept of irrationality. Elsewhere in Indian works however it is stated that various square root values cannot be *exactly determined*, which strongly suggests an initial knowledge of irrationality.

Indeed an early method for calculating square roots can be found in some Sutras, the method involves repeated application of the formula: *A* = (*a*^{2} + *r*) = *a* + *r*/2*a*, *r* being small.

Example 4.2.2: Application of formula for calculating square roots.

If

A=10, takea= 9 andr= 1.

Thus 10 = (32 + 1) = a + r/2a = 3 + 1/6 = 3.16667 in (modern) decimal notation.

10 = 3.162278 to six decimal places when calculated on a calculator. Thus, after only one application of the formula, a moderately accurate value has been calculated.

C Srinivasiengar thus states:

.

..The credit of using irrational numbers for the first time must go to the Indians.[CS, P 15]

Many of the Vedic contributions to mathematics have been neglected or worse. When it first became apparent that there was geometry contained within works that was not of Greek origin, historians and mathematical commentators went to great lengths to try and claim that this geometry was Greek influenced (to a greater or lesser extent).

It is undeniable that none of the methods of Greek geometry are discernible in Vedic geometry, but this merely serves to support arguments that it is independently developed and not in some way borrowed from Greek sources.

In light of recent evidence and more accurate dating it has been even more strongly claimed by A Seidenberg (in S Kak) that:

…Indian geometry and mathematics pre-dates Babylonian and Greek mathematics.[SK1, P 338]

This may be a somewhat extreme standpoint, and it seems likely that there was traffic of ideas in all directions of the Ancient world, but there is little doubt that the vast majority of Indian work is original to its writers. It may lack the cold logic and truly abstract character of modern mathematics but this observation further helps to identify it as uniquely Indian. Of all the mathematics contained in the Vedangas it is the definite appearance of decimal symbols for numerals and a place value system that should perhaps be considered the most phenomenal.

Before the period of the Sulbasutras was at an end the Brahmi numerals had definitely begun to appear (c. 300BC) and the similarity with modern day numerals is clear to see (see Figures 7.1 and 7.3). More importantly even still was the development of the concept of *decimal place value*. M Pandit in a recent paper (discussed in RG2) has shown certain rules given by the famous Indian grammarian Panini (c. 500 BC) imply the concept of the mathematical zero. Further to this there is a small amount of evidence of the use of symbols for numbers even earlier in the Harrapan culture. My evidence comes primarily from a paper by S Kak, which analyses some of Panini‘s work, and there is further support from a paper by S Sinha. B Datta and A Singh also give evidence of an early emergence of numerical forms and the decimal place value system.

## 5: Jainism

Following the decline of the Vedic religion around 400BC, the Jaina religion (and Buddhism) became the prominent religion(s) on the Indian subcontinent and gave rise to Jaina mathematics. N Dwary (and others) contend:

…According to the religious literature of the Janias, the knowledge of “Sankhyana”(i.e., the science of numbers, which included arithmetic and astronomy)was considered to be one of the principal accomplishments of Jain priests.[ND1, P 39]

The main Jaina works on mathematics date from around 300BC to 400AD, but the Jaina religion was in its infancy as far back as 500BC. Further, as a point of slight interest, the Jaina religion has still not completely died out, and up till the 19^{th} century some very minor works were produced.

Jaina mathematics played an important role in bridging the gap between ‘ancient’ Indian mathematics and the so-called ‘Classical period’, which was heralded by the work of Aryabhata I in the 6^{th} century.

Regrettably there are few extant Jaina works, but in the limited material that exists an incredible level of originality is demonstrated. Perhaps the most historically important Jaina contribution to mathematics as a subject is the progression of the subject from purely practical or religious requirements. During the Jaina period mathematics became an abstract discipline to be cultivated “for its own sake”. G Joseph states:

…The Jaina contribution to this change should be recognised.[GJ, P 250]

Despite its important historical position, relatively little attention has been paid to Jaina mathematics, and it remains seemingly discarded by many historians. This however is a massive oversight, because within Jaina works there are many remarkable ‘new’ results, and developments of topics found in Vedic works. It is worth briefly noting here that there is great uncertainty as to whether Vedic works influenced the Jaina mathematicians. Throughout the history of Indian mathematics it seems very possible each ‘leap’ was made without knowledge of previous discoveries. This has been suggested to be primarily due to the size of the sub-continent and imperfect channels of communication within it. Another contribution was the ‘oral transmission’ tradition of Ancient Indian mathematics, which resulted in knowledge being lost over time.

There are several significant Jaina works, including the *Surya Prajinapti* (4^{th} c. BC) and several Sutras. There is also evidence of individual mathematicians including Bhadrabahu (c. 300 BC, possibly lived in the Mysore State), C Srinivasiengar conjectures he wrote two works (which have yet to be unearthed), and Umaswati, a commentator from 2^{nd} century BC, (possibly lived around 150 BC). Umaswati is known as a great writer on Jaina metaphysics but also wrote a work *Tattwarthadhigama-Sutra Bhashya*, which contains mathematics.

Among the important developments of the Jainas are:

Theory of numbers.

There is a great fascination in Jain philosophy with the enumeration of large numbers, selected examples of ‘time periods’ mentioned include 756 1011 8400000028 days, calledshirsa prahelika, and 2588 years.All numbers were classified into three sets:

Enumerable, Innumerable and Infinite.Fivedifferent types of infinity are recognised in Jaina works:

Infinite inoneandtwo directions, infinite inarea, infiniteeverywhereand infiniteperpetually.

This theory is quite incredible and was not realised in Europe until the late 19^{th}century work of George Cantor. Indeed much of the Jaina theory of infinity is extremely advanced for the time in which it was conceived.

Also found in Jaina works:

Knowledge of the fundamental laws of indices.

Arithmetical operations.

Geometry.

Operations with fractions.

Simple equations.

Cubic equations.

Quartic equations (the Jaina contribution to algebra is severely neglected).

Formula for p (root 10, comes up almost inadvertently in a problem about infinity).

Operations with logarithms.

Sequences and progressions.

Finally of interest is the appearance of Permutations and Combinations in Jaina works, which resulted in the formation of an early *Pascal triangle*, called *Meru Prastara*, many centuries before Pascal himself ‘invented’ it. This is another case where Indian contributions have been neglected severely.

Example 5.1: Meru Prastara rule found in Jain works.

Rule is simpler than that of Pascal, and is based on the simple formula:

_{n+1}C_{r}=_{n}C_{r}+_{n}C_{r-1}

Example 5.2: Formulas for permutations and combinations.

Correct formulas for both permutations and combinations are found in Jaina works:

_{n}C_{1}=n,_{n}C_{2}=n(n– 1)/1.2,_{n}C_{3}=n(n– 1)(n– 2)/1.2.3_{n}P_{1}=n,_{n}P_{2}=n(n– 1),_{n}P_{3}=n(n– 1)(n– 2)

The contribution of the Buddhist school should also be briefly discussed. Although in the shadow of Jaina developments, evidence suggests Buddhist scholars were well versed in the use of the decimal place values system and that knowledge of Gainta was considered important.

There was no sudden decline of the Jaina religion as such but from the beginning of the 6^{th} century the work of a mathematician named Aryabhata surpassed all previous work of the Indian sub-continent and brought about the ‘Classical period’ of Indian mathematics, (which lasted 600 years). However prior to discussing the work of Aryabhata there is a major piece of Indian mathematical work yet to be discussed. That is the Bakhshali manuscript.

## 6: The Bakhshali manuscript

The Bakhshali manuscript, which was unearthed in the 19^{th} century, does not appear to belong to any specific period. Although that said, G Joseph classes it as a work of the early ‘classical period’, while E Robertson and J O’Connor suggest it may be a work of Jaina mathematics, and while this is chronologically plausible there is no proof it was composed by Jain scholars. L Gurjar discusses its date in detail, and concludes it can be dated no more accurately than ‘between 2^{nd} century BC and 2^{nd} century AD’. He offers compelling evidence by way of detailed analysis of the contents of the manuscript (originally carried by R Hoernle). His evidence includes the language in which it was written (‘died out’ around 300 AD), discussion of currency found in several problems, and the absence of techniques known to have been developed by the 5^{th} century. Further support of these dates is provided by several occurrences of terminology found only in the manuscript, (which form the basis of a paper by M Channabasappa).

The controversy and debate surrounding the date of the Bakhshali manuscript was particularly intense when it was first discovered and highlights the resistance of European historians to accept new discoveries and evidence of the origins of various mathematical results. Clearly establishing a date for the composition of the manuscript is extremely important as it has a vast bearing on the significance of its mathematical content. This is in fact the primary requirement for accurate dating of all mathematical works.

The first translation of the manuscript was carried out by G R Kaye, however he was quite unscrupulous in his work and attempted to date the manuscript as 12^{th} Century in order to justify Arabic and Greek influences on the text. Kaye even went as far as to question the Indian origin of the manuscript.

However the vast majority of his translation and suggestions made about the origins of the manuscript have been debunked by less biased (and more accurate) translations. G Joseph further criticises Kaye and comments:

…It is particularly unfortunate that Kaye is still quoted as an authority on Indian mathematics.[GJ, Ps 215-216]

To slightly confuse the issue, it is now considered (almost without doubt) that the manuscript found at Bakhshali is a copy (of the original work) dating from around the 8^{th} century, and certainly no later than 950 AD. The scholar R Hoernle was the first to reach this conclusion following detailed analysis of the manuscript.

I am content to agree with the (prominent) historians who have placed the date at pre 450 AD and identified the ‘current’ version as a copy. Avoiding further debate, L Gurjar states that the Bakshali manuscript is the:

…Capstone of the advance of mathematics from the Vedic age up to that period.[LG, P49]

Although, as much work was lost between ‘periods’, we cannot fully gauge continuity of progress and it is possible the composer(s) of the Bakhshali manuscript were not fully aware of earlier works and had to start from ‘scratch’. This would make the work an even more remarkable achievement.

The B. Ms. was written on leaves of birch, in Sarada characters and in Gatha dialect, which is a combination of Sanskrit and Prakrit. This may go some way to explaining the number of inaccurate translations. Many of the historians who have been involved in translating ancient Indian works have done so poorly, due to the obscure script, or alternatively because they have not understood the mathematics fully. More worryingly there could be unscrupulous reasons for poor translating in order to play down the importance of ancient Indian works, because they challenge the Eurocentric ideal.

The B. Ms. highlights developments in Arithmetic and Algebra. The arithmetic contained within the work is of such a high quality that it has been suggested:

…In fact [the] Greeks [are] indebted to India for much of the developments in Arithmetic.[LG, P 53]

This quote ‘throws open’ the traditional Eurosceptic opinion of the history (and origins) of mathematics. Yet even today histories of mathematics rarely acknowledge this contribution of the Indian sub-continent and the B. Ms. is rarely if ever mentioned.

There are eight principal topics ‘discussed’ in the B. Ms:

Examples of the rule of three (and profit and loss and interest).

Solution of linear equations with as many as five unknowns.

The solution of the quadratic equation (development of remarkable quality).

Arithmetic (and geometric) progressions.

Compound Series (some evidence that work begun by Jainas continued).

Quadratic indeterminate equations (origin of typeax/c=y).

Simultaneous equations.

Fractions and other advances in notation including use of zero and negative sign.

Improved method for calculating square root (and hence approximations for irrational numbers). The improved method (shown below) allowed extremely accurate approximations to be calculated:A= (a^{2}+r) =a+r/2a– {(r/2a)^{2}/ 2(a+r/2a)}

Example 6.1: Application of square root formula.

Again we can calculate 10, where a = 3 and r = 1.

10 = (32 + 1) = 3 + 1/6 – {(1/36)/2(3 + 1/6)}

= 3 + 1/6 – {(1/36)/(19/3)}

= 3 + 1/6 – 1/228

= 3.16228… in decimal form

10 = 3.16228 when calculated on a calculator and rounded to five decimal places.

Example 6.2: Quadratic equation as found in B. Ms.

If the equation given is

dn^{2}+ (2a–d)n-2s= 0

Then the solution is found using the equation:n= (- (2a–d) (2a–d)^{2}+8ds))/2d

Which is the quadratic equation witha=d,b= 2a–d, andc= 2s.

Example 6.3: Linear equation with 5 variables.

The following problem is stated in the B. Ms:

“Five merchants together buy a jewel. Its price is equal to half the money possessed by the first together with the money possessed by the others, or one-third the money possessed by the second together with the moneys of the others, or one-fourth the money possessed by the third together with the moneys of the others…etc. Find the price of the jewel and the money possessed by each merchant.Solution.

We have the following systems of equations:x_{1}/2 +x_{2}+x_{3}+x_{4}+x_{5}=px_{1}+x_{2}/3 +x_{3}+x_{4}+x_{5}=px_{1}+x_{2}+x_{3}/4 +x_{4}+x_{5}=px_{1}+x_{2}+x_{3}+x_{4}/5 +x_{5}=px_{1}+x_{2}+x_{3}+x_{4}+x_{5}/6 =p

Then ifx_{1}/2 +x_{2}/3 +x_{3}/4 +x_{4}/5 +x_{5}/6 =qthe equations become (^{377}/_{60})q=p.

A number of possible answers can be obtained. This is the origin of the indeterminate equation of the typeax/c=y, the theory of which was greatly developed, and later perfected by Bhaskara II, four hundred years before it was discovered in Europe.

Ifq= 60 thenp= 377 andx_{1}= 120,x_{2}= 90,x_{3}= 80,x_{4}= 75 andx_{5}= 72

With regard to my central discussion of the neglect of Indian mathematics I feel it is important to highlight the presence of moderately advanced algebra in the B. Ms. Historians of mathematics debate whether true algebra ‘began’ in Greece or Arabia, and little mention is ever made of Indian algebra. In light of my own research I feel that early Arabic algebra (c. 800 AD) in no way *surpasses* the level of understanding of 6^{th} century Indian scholars.

The Bakhshali manuscript is a unique piece of work and while it not only contains mathematics of a remarkably high standard for the time period, also, in contrast to almost all other Indian works composed before and after, the method of the commentary follows a highly systematic order of:

i. Statement of the rule (

sutra)

ii. Statement of the examples (udaharana)

iii. Demonstration of the operation (karana) of the rule.

The work is considerably less concise than other Indian works which were often (if not always) written in a poetic form comprising of short statements of rules, and rarely included examples. This poetic form was favoured, not only as it gave the authors an opportunity to demonstrate their skills but also because of the limited supplies of writing equipment available.

The reasons for the composition of the B. Ms. are unknown but it seems possible that the motivation was to “bring out” the developments of mathematics during the time period. Thus it seems very likely it was composed for solely academic, intellectual and interest ends, in short, mathematics for mathematics sake.

By the end of the 2^{nd} century AD mathematics in India had attained a considerable stature, and had become divorced from purely practical and religious requirements, (although it is worth noting that over the next 1000 years the majority of mathematical developments occurred within works on astronomy). The topics of algebra, arithmetic and geometry had developed significantly and it is widely thought that the decimal place value system of notation had been (generally) perfected by 200 AD, the consequence of which was far reaching.

## 7: Decimal numeration and the place-value system

I have already mentioned on several occasions the development of a decimal place value system of numeration and there is now very little doubt among historians that this invention originated from the Indian subcontinent. That said, it was considered, until recently, that Arabic scholars were responsible for the system, as C Srinivasiengar writes:

…During the earlier decades of this century (20[CS, P 2]^{th}) attempts were made to credit this invention wholly or in part to the Arabs.

Further attempts have been made to attribute the first use of a place value system to the ancient Babylonian civilisation of Mesopotamia. While it cannot be denied that the Babylonians used a place value system, their’s was sexagismal (base 60), and while the concept of place value may have come from Mesopotamia, the Indians were the first to use it with a decimal base (base 10).

All current evidence points towards the Indian system having been influenced by the base 10 Chinese ‘counting boards’ and the place value system of the Babylonians but combined use of decimal numerals and place value first occurred on the Indian subcontinent. Without doubt the use of a decimal base originates from the most basic human instinct of counting on one’s fingers. The key contribution of the Indians however is not in the development of nine (recognisable) symbols to represent the numbers one to nine, but the invention of the place holder *zero*.

The great 18^{th} century European mathematician Laplace best described the ‘invention’ of the decimal place value system as such:

…The idea of expressing all quantities by nine figures whereby is imparted to them both an absolute value and one by position is so simple that this very simplicity is the very reason for our not being sufficiently aware how much admiration it deserves.[CS, P 5]

Beyond not being fully appreciated D Duncan discusses briefly the enduring problem of Eurocentric scholars who long assumed the symbol for zero was a Greek invention, with no proof at all. The claims were based of pure speculations that zero came from the Greek letter omicron (O), the first letter of the Greek word *ouden* meaning empty. We know this to be untrue, but it serves as a timely reminder of the struggle for recognition of Indian mathematical developments.

There is wide ranging debate as to when the decimal place value system was developed, but there is significant evidence that an early system was in use by the inhabitants of the Indus valley by 3000 BC. Excavations at both Harappa and Mohenjo Daro have supported this theory. At this time however a ‘complete’ place value system had not yet been developed and along with symbols for the numbers one through nine, there were also symbols for 10, 20, 100 and so on.

The formation of the numeral forms as we know them now has taken several thousand years, and for quite some time in India there were several different forms. These included Kharosthi and Brahmi numerals, the latter were refined into the Gwalior numerals, which are notably similar to those in use today (see Figure 7.1). Study of the Brahmi numerals has also lent weight to claims that decimal numeration was in use by the Indus civilisation as correlations have been noted between the Indus and Brahmi scripts.

It is uncertain how much longer it took for zero to be invented but there is little doubt that such a symbol was in existence by 500 BC, if not in widespread use. Evidence can be found in the work of the famous Indian grammarian Panini (5^{th} or 6^{th} century BC) and later the work of Pingala a scholar who wrote a work, *Chhandas-Sutra* (c. 200 BC). The first documented evidence of the use of zero for mathematical purposes is not until around 2^{nd} century AD (in the Bakhshali manuscript). The first recorded ‘non-mathematical’ use of zero dates even later, around 680 AD, the number 605 was found on a Khmer inscription in Cambodia. Despite this it seems certain that a symbol was in use prior to that time. B Datta and A Singh discuss the likelihood that the decimal place value system, including zero had been ‘perfected’ by 100 BC or earlier. Although there is no concrete evidence to support their claims, they are established on the very solid basis that new number systems take 800 to 1000 years to become ‘commonly’ used, which the Indian system had done by the 9^{th} century AD.

The inventor of the zero symbol is unknown, but what is known is that it was firstly denoted by a dot, then possibly a circle with a dot in the centre, and later by the oval shape we now use. Prior to its invention, Indian mathematicians had already taken to leaving an empty column on their counting boards and clearly at some point this empty space was filled. The Indians referred to zero as ‘sunya’ meaning void. Again, although evidence points towards a Mesopotamian origin for a place holder, their ‘zero’ (two slanted bars) was not used in conjunction with a decimal base.

Having become firmly established in academic circles in India by the 6^{th} century, the decimal place value system spread across the world. Initially to China and Alexandria, then to the Arab empire where it became the system of choice of the scholars in Baghdad by the 8^{th} century.

Arabic scholars during this time improved the system by introducing decimal fractions. The system also spread into Spain, as has been previously discussed southern Spain was under Arabic rule into the 12^{th} century. It took much longer for the system to be accepted in mainland Europe, but eventually by the 16^{th} century it was widely used. That said, both prejudice and suspicion continued to be widespread, while orthodoxy also played its part in the continued use of Roman numerals. The last significant case of an attempt to abolish the Indian decimal place value system was in Sweden in the early 18^{th} century.

This is clearly a very brief overview of the phenomenal development of the decimal place value system, without which it is accepted ‘higher mathematics’ would not be possible. It is impossible for me to do justice to its importance in such few words, so I will conclude with a quote from G Halstead who commented:

…The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habituation and a name, a picture, a symbol but helpful power, is the characteristic of the Hindu race from whence it sprang. No single mathematical creation has been more potent for the general on go of intelligence and power.[CS, P 5]

Figure 7.1: Indian numeral forms. [GJ, P 241]

Figure 7.2: Numeral forms found in Bakhshali Manuscript, showing place value and use of zero. [GJ, P 241]

Figure 7.3: Brahmi numerals. [DD, P 163]

Figure 7.4: Progression of Brahmi number forms through the centuries (column far left showing forms in use by 500 AD). [AS/BD (vol. 1), P 120]

Figure 7.5: Numerical forms (including zero) by found in 20^{th} century Indian texts. [AS/BD (vol. 1), P 121]

## 8: The Classical period: I Introduction

Following the Bakhshali manuscript (assuming it was composed no later than 200 AD) Indian mathematics suffered a slump, described by L Gurjar as:

…A deep and dire degeneration.[LG, P 78]

This was in part due to massive communication problems, but also undoubtedly to the huge political upheaval that took place between the 2^{nd} and 4^{th} centuries AD, prior to the capturing of power of most of India by the Imperial Guptas.

During this period however the practice of writing *Siddhantas* (astronomical works), which had started around 500 BC, continued. Of the Siddhantas the *Pitamaha Siddhanta* is the oldest (c. 500 BC) and the *Surya Siddhanta *(c. 400 AD, author unknown, influenced Aryabhata) is the best known. The major contribution of these works was the invention of the sine function. R Gupta comments:

…Trigonometry based on sine (instead of chord (as in Greek works)) and related functions is another important gift of India to the world of mathematics.[RG2, P 134]

Following the establishment of the ‘Gupta dynasty’ a period of political stability arose, which doubtlessly contributed to the ascent of a “galaxy” of mathematician-astronomers, led by Aryabhata. These men were, first and foremost, astronomers, but the mathematical requirements of astronomy (and no doubt further interest) led to them developing many areas of mathematics, as such, I will refer to them just as mathematicians. The vast majority of the works of the ‘classical period’ were however, in effect, Siddhantas.

## 8 II. Aryabhata and his commentators

Aryabhata, who is occasionally known as Aryabhata I, or Aryabhata the elder to distinguish him from a tenth century astronomer of the same name, stands as a pioneer of the revival of Indian mathematics, and the so called *‘classical period*‘, or ‘Golden era’ of Indian mathematics. Arguably the Classical period continued until the 12^{th} century, although in some respects it was over before Aryabhata‘s death following a costly, if ultimately successful, war with invading Huns which resulted in the eroding of the Gupta culture (D Duncan P 171). As mentioned, the classical period arose following a ‘dark period’ of significant political instability 200-400 AD, which caused the widespread stagnation of mathematical development.

We can accurately claim that Aryabhata was born in 476 AD, as he writes that he was 23 years old when he wrote his most significant mathematical work the *Aryabhatiya* (or *Arya Bhateeya*) in 499 AD. He was a member of the Kusuma Pura School, but is thought to have been a native of Kerala (in the extreme south of India), although unsurprisingly there is some debate. Further debate surrounds how important the work of Aryabhata actually was. In light of discoveries of both Vedic and Jaina mathematics it has been suggested that his work is of less significance mathematically. However as Gurjar claims it is more than likely that the majority, if not all prior mathematical work may not have been known to him, which makes his contribution remarkable. Either way, Aryabhata‘s work was to have massive influence on those who followed, in a similar way that the work of Luca Pacioli influenced the Italian renaissance mathematicians in the late 15^{th} century.

As previously mentioned the only extant work of Aryabhata is his key work, the *Aryabhatiya*, a concise astronomical treatise of 118 verses written in a poetic form, of which 33 verses are concerned with mathematical rules. It is important here to point out that no proofs are contained with his rules, and this is perhaps a primary reason for the neglect by western scholars. As Indian mathematics is (generally) devoid of proof it is not considered ‘true’ mathematics in its purest sense. However I believe that adopting this stance is to deny the very origin of remarkable discoveries in mathematics, which may well have been the aim of Eurocentric scholars, as it allowed them to neglect the importance of Indian works in favour of European works.

In the mathematical verses of the *Aryabhatiya* the following topics are covered:

Arithmetic:

Method of inversion.

Various arithmetical operators, including the cube and cube root are though to have originated in Aryabhata’s work. Aryabhata can also reliably be attributed with credit for using the relatively ‘new’ functions of squaring and square rooting.Algebra:

Formulas for finding the sum of several types of series.

Rules for finding the number of terms of an arithmetical progression.

Rule of three – improvement on Bakshali Manuscript.

Rules for solving examples on interest – which led to the quadratic equation, it is clear that Aryabhata knew the solution of a quadratic equation.Trigonometry:

Tables of sines,notcopied from Greek work (see Figure 8.2.1).

Gupta comments:

…The Aryabhatiya is the first historical work of the dated type, which definitely uses some of these (trigonometric) functions and contains a table of sines.[RG3, P 72]Spherical trigonometry (some incorrect).

Geometry:

Area of a triangle, similar triangles, volume rules.

It has been suggested that Aryabhata‘s geometry was borrowed from the Jaina works, but this seems unlikely as it is generally accepted he would not have been familiar with them.

Also of relevance is the use of ‘word numerals’ and ‘alphabet numerals’, which are first found in Aryabhata‘s work. We can argue that this was not due to the absence of a satisfactory system of numeration but because it was helpful in poetry. C Srinivasiengar quaintly describes it as an:

…Exceedingly queer, if original method of enumeration.[CS, P 43]

However the work of Aryabhata also affords a proof that:

…The decimal system was well in vogue.[CS, P 43]

Of the mathematics contained within the *Aryabhatiya* the most remarkable is an approximation for p, which is surprisingly accurate. The value given is: p = 3.1416

With little doubt this is the most accurate approximation that had been given up to this point in the history of mathematics. Aryabhata found it from the circle with circumference 62832 and diameter 20000. Critics have tried to suggest that this approximation is of Greek origin. However with confidence it can be argued that the Greeks only used p = 10 and p = ^{22}/_{7} and that no other values can be found in Greek texts.

I note with slight concern for the strength of my last comment that the Egyptian scholar Claudius Ptolemy derived the same value 300 years earlier, although there is no suggestion of a link between these two cases. Further to deriving this highly accurate value for p, Aryabhata also appeared to be aware that it was an ‘irrational’ number and that his value was an approximation, which shows incredible insight. Thus even accepting that Ptolemy discovered the 4 decimal place value, there is no evidence that he was aware of the concept of irrationality, which is extremely important. Inexplicably Aryabhata preferred to use the approximation p = 10 (= 3.1622) in practice!

Aryabhata‘s work on astronomy was also pioneering, and was far less tinged with a mythological flavour. Among many theories he was the first to suggest that diurnal motion of the ‘heavens’ is due to rotation of the earth about its axis, which is incredibly insightful (unsurprisingly he was criticised for this.)

In the field of ‘pure’ mathematics his most significant contribution was his solution to the indeterminate equation: *ax* – *by* = *c*

R Gupta states:

…Aryabhata (also) made notable contributions to algebra[RG3, P73]

Although Indian mathematics has become often ignored this was not always the case. The *Aryabhatiya* was translated into Arabic by Abu’l Hassan al-Ahwazi (before 1000 AD) as *Zij al-Arjabhar* and it is partly through this translation that Indian computational and mathematical methods were introduced to the Arabs, which will have had a significant effect on the forward progress made by mathematics. The historian A Cajori even goes as far as to suggest that:

…Diophantus

, the father of Greek algebra, got the first algebraic knowledge from India.[RG4, P 12]

This theory is supported by evidence that the eminent Greek mathematician Pythagoras visited India, which further ‘throws open’ the Eurocentric ideal.

Example 8.2.1: Solution of 137*x* + 10 = 60*y*, as found in the *Aryabhatiya*.

The general solution is found as follows:

137x+ 10 = 60y60) 137 (2 (60 divides into 137 twice with remainder 17, etc)12017( 60 ( 3519) 17 ) 198 ) 9 (181The following column of remainders, known as

valli(vertical line) form is constructed:2

3

1

1The number of quotients, omitting the first one is 3. Hence we choose a multiplier such that on multiplication by the last residue, 1 (in red above), and subtracting 10 from the product the result is divisible by the penultimate remainder, 8 (in blue above). We have 1 18 – 10 = 1 8. We then form the following table:

2 2 2 2 297 3 3 3 130 130 1 1 37 37 1 19 19 The multiplier 18 18 Quotient obtained 1This can be explained as such: The number 18, and the number above it in the first column, multiplied and added to the number below it, gives the last but one number in the second column. Thus, 18 1 + 1 = 19. The same process is applied to the second column, giving the third column, that is, 19 1 + 18 = 37. Similarly 37 3 + 19 = 130, 130 2 + 37 = 297.

Then

x= 130,y= 297 are solutions of the given equation. Noting that 297 = 23 (mod 137) and 130 = 10 (mod 60), we getx= 10 andy= 23 as simple solutions. The general solution isx= 10 + 60m,y= 23 + 137m. If we stop with the remainder 8 in the process of division above then we can at once getx= 10 andy= 23. (Working omitted for sake of brevity).

This method was calledKuttaka, which literally means pulveriser, on account of the process of continued division that is carried out to obtain the solution.

Figure 8.2.1: Table of sines as found in the *Aryabhatiya*. [CS, P 48]

The work of Aryabhata was also extremely influential in India and many commentaries were written on his work (especially his Aryabhatiya). Among the most influential commentators were:

Bhaskara I (c 600-680 AD) also a prominent astronomer, his work in that area gave rise to an extremely accurate approximation for the sine function. His commentary of the

Aryabhatiyais of only the mathematics sections, and he develops several of the ideas contained within. Perhaps his most important contribution was that which he made to the topic of algebra.

Lalla (c 720-790 AD) followed Aryabhata but in fact disagreed with much of his astronomical work. Of note was his use of Aryabhata‘s improved approximation of pi to the fourth decimal place. Lalla also composed a commentary on Brahmagupta‘sKhandakhadyaka.

Govindasvami (c 800-860 AD) his most important work was a commentary on Bhaskara I‘s astronomical workMahabhaskariya, he also considered Aryabhata‘s sine tables and constructed a table which led to improved values.

Sankara Narayana (c 840-900 AD) wrote a commentary on Bhaskara I’s work Laghubhaskariya (which in turn was based on the work of Aryabhata). Of note is his work on solving first order indeterminate equations, and also his use of the alternate ‘katapayadi’ numeration system (as well as Sanskrit place value numerals)

Following Aryabhata‘s death around 550 AD the work of Brahmagupta resulted in Indian mathematics attaining an even greater level of perfection. Between these two ‘greats’ of the classic period lived Yativrsabha, a little known Jain scholar, his work, primarily *Tiloyapannatti*, mainly concerned itself with various concepts of Jaina cosmology, and is worthy of minor note as it contained interesting considerations of infinity.

## 8 III. Brahmagupta, and the influence on Arabia

Brahmagupta was born in 598 AD, possibly in Ujjain (possibly a native of Sind) and was the most influential and celebrated mathematician of the Ujjain school.

It is important here to note that one must not ignore contributions made by Varahamihira, who was an influential figure at the same Ujjain school during the 6^{th} century. He is thought to have lived from 505 AD till 587 AD and made only fairly small contributions to the field of mathematics, he is described by Ifrah as:*…One of the most famous astrologers in Indian history. *[EFR/JJO’C18, P 1]

However he increased the stature of the Ujjain school while working there, a legacy that was to last for a long period, and although his contributions to mathematics were small they were of some importance. They included several trigonometric formulas, improvement of Aryabhata‘s sine tables, and derivation of the *Pascal triangle* by investigating the problem of computing binomial coefficients.

Returning to Brahmagupta, he not only elaborated the mathematical results of Aryabhata but also made notable contributions to many topics.

L Gurjar describes his results as:

.

..Unique in the history of world mathematics.[LG, P 91]

His contributions to mathematics are found in two works, the first of which *Brahmasphutasiddhanta* (BSS) must be considered one of the important mathematical works from this early period, not only of India, but also of the world. Not only was its mathematical content of an exceptional quality, but the work also had a significant influence on the burgeoning scientific awakening in the Arab empire.

I believe that the Indian influence on Arabic work is often ignored or played down and consider this to be unfortunate (at the least). This issue is definitely worthy of discussion as it is noticeable that much is made of the Greek influence on Arabic works but far less of the Indian influence, which in retrospect was quite significant.

His second work was written much later in his life in 665 AD and was titled *Khandakhayaka*. Although the BSS contains 25 chapters it is generally considered that the first ten chapters make up the first work, and that at a later date Brahmagupta made revisions and additions.

In the BSS among the major developments are those in the areas of:

Arithmetic:

Brahmagupta possessed a greater understanding of the number system (and place value system) than anyone to that point. Many rules are given and an advanced technique for multiplication exhibited.

Operations with zero, Brahmagupta was the first to attempt to divide by zero, and while his attempts; showingn/0 = , were not ultimately successful they demonstrate an advanced understanding of an extremely abstract concept.

Operations with negative numbers.

Theory of Arithmetic progressions.

Algorithm for calculating square roots that is equivalent to the Newton-Raphson iterative formula, but clearly pre-dates it by many centuries. (See chapter 8.6)

Geometry:

Brahmagupta stands in high esteem for his contributions to this topic. A rule that he gives for finding the values of the diagonals of cyclic quadrilaterals is generally known as “Ptolemy’s theorem”. Ptolemy ‘pre-dated’ Brahmagupta by 500 years, so it is wholly reasonable to attribute the ‘discovery’ of these rules to him. However, Brahmagupta‘s independent discovery should still be considered a remarkable achievement.

Furthermore, some of his work (regarding right angled triangles, which was later developed by Mahavira, Bhaskara II, et al) is often attributed to Fibonacci (13^{th}c.) and Vieta (16^{th}c.), highlighting the constant European bias.As L Gurjar quotes:

…(Brahmagupta) derived certain results, which were troubling the brains of Western mathematicians as late as the 17[LG, P 91]^{th}century.

Algebra:

Solutions toNx^{2}+ 1 =y^{2},Pell’s equation, his most outstanding contribution to mathematics. (See chapter 8.6)

He also made many other contributions to solving a variety of algebraic equations, includingax+c=by(which is the focus of a paper by P Majumdar).

Brahmagupta may have been one of the first mathematicians to recognise that the quadratic equation hastwosolutions.

In his other work, the content is far more ‘pure’ astronomy, but an interpolation formula used to calculate values of sines bears great similarity to the Newton–Stirling interpolation formula, which is clearly of great historical and mathematical interest.

Without a doubt, Brahmagupta made remarkable contributions to mathematics (and astronomy) and his work continued to be influential for many centuries. In 860 AD an extensive and important commentary on the BSS was written by Prthudakasvami (or Prithudaka Swami). His work was extremely elaborate and unlike many Indian works did not ‘suffer’ brevity of expression.

The spread of Buddhism (around 500 AD) into China resulted in a period of cultural and scientific exchanges lasting several centuries. Chinese scholars are known to have translated the work of Brahmagupta; this highlights not only the quality of the work but the influence it had on the world outside India. R Gupta mentions four ‘*Brahminical*‘ translations in a paper. During this time the decimal system and notation was adopted by Chinese scholars and as R Gupta states:

…Indian mathematical astronomy exerted a great influence in China during the(glorious)Thang Period(618-907). [RG4, P 11]

The lasting legacy of the BSS however was its translation by *Arab* scholars and its contribution to the ‘forward progress’ of mathematics. These translations, along with translated work of Aryabhata and (possibly) the *Surya Siddhanta *were responsible for alerting the Arabs, and the West to Indian mathematics (and astronomy), as G Joseph states:

…This was to have momentous consequences for the development of the two subjects.[GJ, P 267]

Of particular interest is the well told story of the Indian scholar who traveled to Baghdad, at the behest of Caliph al-Mansur (early ruler of the Arab Empire). R Gupta reports the story as such:

…In the year156 (772/773 AD)there came to Caliph al-Mansur a man(an Ujjain scholar by the name of Kanka)from India, an expert in hisab(computation)bringing with him a work called Sindhind (i.e. Siddhanta) concerning the motions of the planets.[RG, P 12]

A translation of this work, thought to be Brahmagupta‘s BSS, was subsequently carried out by al-Fazari (and an Indian scholar) and had a far-reaching influence on subsequent Arabic works. The famous Arabic scholar al-Khwarizmi (credited with ‘inventing’ algebra) is known to have made use of the translation, called *Zij al-Sindhind*. Al-Khwarizmi (c. 780-850 AD) is known to have written two subsequent works, one based on Indian astronomy (Zij) and the other on arithmetic (possibly *Kitab al-Adad al-Hindi*). Later Latin translations of this second work (*Algorithmi De Numero Indorum*), composed in Spain around the 11^{th} century, are thought to have played a crucial role in introducing the Indian place-value system numerals and the corresponding computational methods into (wider) Europe. Both Indian astronomy and arithmetic had a huge impact in Spain.

This discussion helps to highlight the influence that Indian mathematics had on Arabic mathematics, and ultimately, through Latin translations, on European mathematics, an influence that is considerably neglected. It must be argued that sufficient credit has not been given.

There seems to have been relatively little further ‘interaction’ between Indian and Arab scholars, and thus Indian works had limited, if any, further influence on mathematical developments in other countries. However Indian mathematics continued to flourish independently throughout the subcontinent for another 400 years, and some of the most outstanding contributions to the history of world mathematics were in fact made during this time period.

## 8 IV. Mathematics over the next 400 years (700AD-1100AD)

Mahavira (or Mahaviracharya), a Jain by religion, is the most celebrated Indian mathematician of the 9^{th} century. His major work *Ganitasar Sangraha* was written around 850 AD and is considered ‘brilliant’. It was widely known in the South of India and written in Sanskrit due to his Jaina ‘faith’. In the 11^{th} century its influence was still being felt when it was translated into Telegu (a regional language of the south). Mahavira was aware of the works of Jaina mathematicians and also the works of Aryabhata (and commentators) and Brahmagupta, and refined and improved much of their work. What makes Mahavira unique is that he was not an astronomer, his work was confined solely to mathematics and he stands almost entirely alone in the history of Indian mathematics (at least up to the 14^{th} century) in this respect. He was a member of the mathematical school at Mysore in the south of India and his major contributions to mathematics include:

Arithmetic:

GSS was the first text on arithmetic in the present form. He made the classification of arithmetical operators simpler. Detailed operations with fractions (and unit fractions), but no section on decimals (which were not an Indian invention).

Geometric progressions – he gave almost all required formulae.

Permutations and Combinations:

Extension and systemisation of Jain works. First to give general formula.

Geometry:

Repeated Brahmagupta‘s construction for cyclic quadrilaterals.

Definitions for most geometric shapes.

Algebra:

Work on quadratic, indeterminate and simultaneous equations. Mahavira demonstrated definite understanding of the concept of a quadratic equation having two roots.

Ellipse:

Only Indian mathematician to refer to the ellipse, indeed Indian mathematicians did not study conic sections or anything along these lines. Gave incorrect identity for area of ellipse. His formula for the perimeter of an ellipse is worth noting.

Mahavira’s work, GSS, could be criticised for being nothing more than an extensive commentary on Jaina works, and the work of Aryabhata, Brahmagupta (and Bhaskara I). C Srinivasiengar describes his work as containing:

…(

No)profoundly fundamental discoveries.[CS, P 70]

To some extent this may be true, but it is also unfair. Mahavira made many subtle contributions and elaborated and revised much of the work of previous mathematicians. Furthermore GSS contained many examples to illustrate his rules, unlike many Indian mathematical works.

The influence of his work (within India) must also not be ignored. The very fact it was still in use more than 250 years after it was written is testimony to its importance as a mathematical work.

Example 8.4.1: General formula for combinations, as given by Mahavira.

_{n}C_{r}= {n(n– 1)(n– 2)×××(n–r+ 1)}/1.2.3. … .r

Following Mahavira the most notable mathematician was Prthudakasvami (c. 830-890 AD) a prominent Indian algebraist, who is described by E Robertson and J O’Connor as being:

…Best known for his work on solving equations.[EFR/JJO’C25, P 1]

He also wrote a commentary on Brahmagupta‘s *Brahma Sputa Siddhanta*.

In the early 10^{th} century a mathematician by the name of Sridhara (c. 870-930 AD) may have lived, however there is much debate surrounding his birth and some authors place him as having lived in the 8^{th} century (750 AD). However beyond debate is the fact that he wrote *Patiganita Sara* a work on arithmetic and mensuration. It contained exactly 300 verses and is hence also known by the name *Trisatika*. It includes contributions to the following topics:

Rules on extracting square and cube roots, fractions and eight rules for operations involving zero (not division).

Theory of cyclic quadrilaterals with rational sides.

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

Methods for summation of different arithmetic and geometric series. These methods became standard references in later works.

It is thought that Sridhara also composed a text on algebra, which is now lost, and several other works have been attributed to Sridhara, but there is no certainty if they were indeed written by him. The legacy of Sridhara‘s work was that it had some influence on the work of Bhaskaracharya II, regarded by many as the greatest Indian mathematician.

Prior to Bhaskara II it is worth noting the contributions of Aryabhata II (c 920-1000 AD) a mathematical-astronomer who notably made important contributions to algebra. In his work *Mahasiddhanta* he gives twenty verses of detailed rules for solving *by* = *ax* + *c* (and variations of this equation). Also of note, Vijayanandi (c 940-1010 AD) who made several contributions to trigonometry in the course of his astronomical works, and Sripati.

Sripati (c. 1019-1066 AD, the birth year of 1019 is known to be correct) was a follower of the teachings of Lalla and in fact the most important Indian mathematician of the 11^{th} century. He was the author of several astronomical works, including *Siddhantasekhara*, which contained two chapters devoted to mathematics. His major contributions were in the fields of arithmetic and algebra. His algebra is of particular note; his work includes rules for solving the quadratic equation and simultaneous indeterminate equations.

Further he impressively gave the identity:

(

x+y) = ([x+ (x^{2}–y)]/2) + ([x– (x^{2}–y)]/2)

{Seex= 2,y= 4 2 = 2}

## 8 V. Bhaskaracharya II

Bhaskaracharya, or Bhaskara II, is regarded almost without question as the greatest Hindu mathematician of all time and his contribution to not just Indian, but world mathematics is undeniable. As L Gurjar states:

…Because of his work India gave a definite ‘quota’ to the forward world march of the science.[LG, P 104]

Born in 1114 AD (in Vijayapura, he belonged to Bijjada Bida) he became head of the Ujjain school of mathematical astronomy (Varahamihira and Brahmagupta had helped to found this school or at least ‘build it up’). There is some confusion amongst the texts I have referred to as to the works that he wrote. C Srinivasiengar claims he wrote *Siddhanta Siromani* in 1150 AD, which contained four sections:

1)

Lilavati(arithmetic)

2)Bijaganita(algebra)

3)Goladhyaya(sphere/celestial globe)

4)Grahaganita(mathematics of the planets)

E Robertson and J O’Connor claim that he wrote 6 works, 1), 2) and SS (which contained two sections) and three further astronomical works, including two commentaries on the SS.

G Joseph claims his mathematically significant works were 1), 2), and SS (which indeed he wrote in 1150 and is a highly influential astronomical work). S Sinha however agrees with C Srinivasiengar that *Lilavati* was a section (chapter) of the SS, and thus I will agree with the respected Indian historians.

Lilavati (or Leelavati, there is a charming if unlikely story regarding the origin of the name of this work) is divided into 13 chapters (possibly by later scribes) and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include:

– Definitions.

– Properties of zero (including division).

– Further extensive numerical work, including use of negative numbers and surds.

– Estimation of p.

– Arithmetical terms, methods of multiplication, squaring, inverse rule of three, plus rules of 5, 7 and 9.

– Problems involving interest.

– Arithmetical and geometrical progressions.

– Plane geometry.

– Solid geometry.

– Combinations.

– Indeterminate equations (Kuttaka), integer solutions (first and second order) His contributions to this topic are among his most important, the rules he gives are (in effect) the same as those given by the renaissance European mathematicians (17^{th}Century) yet his work was of 12^{th}Century. Method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.

– Shadow of the gnomon.

The *Lilivati* is written in poetic form with a prose commentary and Bhaskara acknowledges that he has condensed the works of Brahmagupta, Sridhara (and Padmanabha).

However his work is outstanding for its systemisation, improved methods and the new topics that he has introduced. Furthermore the *Lilavati* contained excellent recreative problems and it is thought that Bhaskara’s intention may have been that a student of ‘*Lilavati*‘ should concern himself with the mechanical application of the method. A student of ‘*Bijaganita*‘ should however concern himself with the theory underlying the method.

His work *Bijaganita* is effectively a treatise on algebra and contains the following topics:

– Positive and negative numbers.

– Zero.

– The ‘unknown’.

– Surds.

– Kuttaka.

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

– Simple equations with more than one unknown.

– Indeterminate quadratic equations (of the typeax^{2}+b=y^{2}).

– Quadratic equations.

– Quadratic equations with more than one unknown.

– Operations with products of several unknowns.

Bhaskara derived a cyclic, ‘*Cakraval*‘ method for solving equations of the form *ax*^{2} + *bx* + *c* = *y*, which is usually attributed to William Brouncker who ‘rediscovered’ it around 1657. Bhaskara‘s method for finding the solutions of the problem *Nx*^{2} + 1 = *y*^{2} (so called “Pell’s equation”) is of considerable interest and importance.

His work the *Siddhanta Siromani* is an astronomical treatise and contains many theories not found in earlier works. There is not a large mathematical content, but of particular interest are several results in trigonometry and *calculus* that are found in the work. These include results of differential and integral calculus.

Bhaskara is though to be the first to show that:

sin

x= cosxx

Evidence suggests Bhaskara was fully acquainted with the principle of differential calculus, and that his researches were in no way inferior to Newton‘s, asides the fact that it seems he did not understand the utility of his researches, and thus historians of mathematics generally neglect his outstanding achievement, which is extremely regrettable. Bhaskara also goes deeper into the ‘differential calculus’ and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ‘infinitesimals’

He also gives the (now) well known results for sin(*a* + *b*) and sin(*a* – *b*). There is also evidence of an early form of Rolle‘s theorem;

iff(a) =f(b) = 0thenf‘(x) = 0forsomexwitha<x<b,

in Bhaskara’s work.

There have been several unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara‘s work, but this once again seems like an attempt by European scholars to claim European influence on (all) the great works of mathematics. These claims should be ignored. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians.

## 8 VI. Pell’s equation

Although there were great advancements in many areas of algebra, and the solution of many different forms of equations, from simple forms such as *ax* + *c* = *by*, to forms as complex as *ax*^{2} + *bxy* + *cy*^{2} = *z*^{2} , I have chosen to look in slightly more detail at solution(s) to the so called Pell’s equation, *Nx*^{2} + 1 = *y*^{2}. The equation was so called due to a mistake on the part of Euler, who attributed the solution of the equation to John Pell, a 17^{th} century English scholar, who actually only referred to the equation in a text he wrote on algebra.

The equation was nearly solved by Brahmagupta (628 AD) and the solution was improved by Bhaskara II (1150 AD), leading some historians, including C Srinivasiengar, to suggest:

…It is therefore fitting that this equation be called the Brahmagupta-Bhaskara equation.[CS, P 110]

The complete theory underlying the solution was expounded by Lagrange in 1767, and rests on the theory of continued fractions. It must be briefly noted how remarkable the achievements of Indian scholars were, given the time period in which equations of the Pell’s type were studied. The Indian method involves an element of trail-process but contains no mention of continued fractions. Further to solving equations of the Pell’s type to obtain solutions for the unknowns, Brahmagupta extended his method of solution to find square roots. This contribution is of huge interest as it is essentially the same method rediscovered and used by Newton and Raphson around 1690, which is known as the Newton-Raphson iterative method. Contained within this brief discussion is a small computer code for the *Maple* mathematics package, which uses the Brahmagupta type solution of Pell’s equation to derive extremely accurate square roots.

The Pell’s type of equation was known in India as *Varga Prakriti*, or “equation of the multiplied square”, where prakriti means coefficient and refers to the coefficient *N* (where *N* is a positive integer). As previously mentioned, Bhaskara developed a Chakravala or cyclic method of solution.

Regrettably, due to constraints of space, I will have to forego discussion of the general solutions derived by Brahmagupta and Bhaskara, and will include only an example to illustrate Bhaskara‘s improved ‘cyclic’ method. The following example is of great historical interest. It is found in the Bijaganita of Bhaskara and is also of the form of a problem Fermat set as a problem to fellow mathematician Frenicle in 1657. The smallest solution for *x* and *y* runs into 4 and 5 digits respectively. The *chakravala* method is remarkable, as it requires just a few ‘easy’ steps, while Lagrange‘s solution required complex use continued fractions.

Example 8.6.1: Solution of 67*x*^{2} + 1 = *y*^{2}.

67

x^{2}+ 1 =y^{2}.

Firstly the auxiliary equation 67 12 – 3 = 82 is taken.

Then using Bhaskara’s lemma, whereNa^{2}+k=b^{2}, wherea,b,kare the integers (1, -3 and 8) in the auxiliary equation above, (kbeing positive or negative) then:

N((am+b)/k)^{2}+ ((m^{2}–N)/k) = ((bm+Na)/k)^{2}Thus:

67((1

m+ 8)/-3)^{2}+ ((m2 – 67)/-3) = ((8m+ 67 1)/-3)^{2}(1)Then, by the method of

Kuttakathe solution of (m+ 8)/-3 = an integer, ism= -3t+ 1.

Puttingt= -2, we getm= 7, which makes [m^{2}– 67] least.

On substituting this value, the equation(1)reduces to:

67 52 + 6 = 412

Again, by the lemma:67((5

n+ 41)/6)^{2}+ ((n^{2}-67)/6) = ((41n+ 67 5)/6)^{2}(2)Then the solution of (5

n+ 41)/6 = a whole number, isn= 6t+ 5. [n^{2}– 67] will be least for the valuet= 0, that is, whenn= 5. The equation(1)then becomes:67 112 – 7 = 902

Now we form:

67((11

p+ 90)/-7)^{2}+ ((p^{2}– 67)/-7) = ((90p+ 67 11/-7)^{2}(3)The solution of (11

p+ 90)/-7 = an integral number, isp= -7t+ 2. Takingt= -1, we havep= 9; and this value makes [p^{2}– 67] least. Substituting that into(3)we get:67 272 – 2 = 2212

By the Principle of Composition of Equals, we get from the above equation:

67(2.27.221)

^{2}+ 4 = (2212 + 67 272)^{2}

Or 67(11934)^{2}+ 4 = (97684)^{2}

Dividing out by 4 we have:

67(5967)

^{2}+ 1 = (48842)^{2}

Hence *x* = 5967, *y* = 48842 is a solution of the equation 67*x*^{2} + 1 = *y*^{2}. (See C Srinivasiengar Ps 110-133 for further details on Indian solutions to Pell’s equation.)

I will conclude my brief discussion of Pell’s equation by illustrating Brahmagupta‘s method for calculating square roots with the following example. The method was used by Brahmagupta to find the square root of the integer *N*, when *Nx*^{2} + 1 = *y*^{2}, and solutions for *x* and *y* are known.

Example 8.6.2: Finding the square root of *N*, from *Nx*^{2} + 1 = *y*^{2}

If

N=5, theny^{2}= 1 + 5x^{2}. It can be observed that 5 = (y^{2}– 1)/x^{2}and that (y^{2}– 1)/x^{2}»y^{2}/x^{2}.

This is the key to finding the square root of 5, as 5 =y/x. With ease we can identifyy= 9 andx= 4 as solutions to this equation, and we see 5 » 9/4 = 2.25, (5 = 2.236067978…).

Clearly the largeryandxare, the better the approximation is. This can shown using the followingMapleprogramme:> n:=5:

> f:=(x,y)->(2*x*y,y*y+n*x*x):

> m:=0:

> x:=4

> y=:9

> while m 5 do

> m:=m+1;

> print(x,y,evalf(y/x,20), evalf(y/x-sqrt(n),50));

> a:=f(x,y);

> x:=a[1];

> y:=a[2];

> end do:The output gives the following:

4, 9 (solutions ofxandy)

2.25000000000000000000 (y/xto 20 decimal places)

0.0139320225002103035908263312687237645593816403885 (error betweeny/xand 5)

By the 5^{th}step the following result is given:

25840354427429161536, 57780789062419261441 (5^{th}pair of solutions forxandy)

2.2360679774997896964 (y/xto 20 d.p.s)0.3348791201 10

^{-39}(error)This result is extremely accurate, and the method must be considered brilliant given it was derived by Brahmagupta in 628 AD.

## 8 VII. The end of the Classic period

The work of Bhaskara was considered the highest point Indian mathematics attained, and it was long considered that Indian mathematics ceased after that point. Extreme political turmoil through much of the sub-continent shattered the atmosphere of discovery and learning and led to the stagnation of mathematical developments as scholars contented themselves with duplicating earlier works.

Recent discoveries however have found that, despite political turmoil, mathematics continued to a high degree in the south of India up to the 16^{th} century. The South of India avoided the worst of the political upheavals of the subcontinent, and the Kerala School of mathematics flourished for some time, producing some truly remarkable results. These results, the most notable of which are in the field of infinite series expansions of trigonometric functions, are generally inaccurately attributed to great European mathematicians of the 18^{th} century including Newton, Leibniz and Gregory. However, slowly, this rigid position is shifting somewhat.

Before going on to discuss the Kerala contribution to mathematics it is worth noting that by the time of Bhaskara II‘s death Indian mathematics of the 5^{th} and 6^{th} centuries had exerted a significant influence on mathematics across the world. By the 11^{th} century a number of important Arabic works had been written, based on translations of a number of Indian astronomical works. As the Arab Empire stretched as far as southern Spain, much of this work based on Indian science made its way into southern Europe and was subsequently translated into Latin.

Sadly there is very little recognition of these facts, and even though the Arabic (and hence some Indian) works were prevalent in Spain, they did not transmit any further into Europe, which was still to fully ‘awaken and probably ‘resisted’ the works, and many were subsequently lost. However ultimately a few Latin translations of Indo-Arabic works did flow into wider Europe, causing a step towards the renaissance.

This brief return to this discussion is primarily to serve as a reminder that Indian mathematics has had a far greater influence on the forward progress of mathematics (in conjunction with enlightened Arab scholars, and ultimately a handful of pioneering European scholars) than is generally mentioned. A prime example is the work of Fibonacci, which shows appreciation of Indo-Arabic work as early as the 12th century. In short, around the time of Bhaskara’s death (12th c.) Indian mathematics (of the 5^{th}-7^{th} centuries) was still exerting a significant influence throughout the world.

As mentioned, it was long considered that following the ‘high point’ of the work of Bhaskara that Indian mathematics fell into a steep decline. To some extent this is true, only shortly after Bhaskara‘s death India was engulfed in war (Mongol invasions) and political turmoil. Consequently the atmosphere of security and tranquillity was lost, which was doubtlessly a major contributory factor to the barrenness of scientific activity and achievement. As C Srinivasiengar quotes:

…India suddenly fell into a state of “torpor” and never recovered from this torpor until the advent of mathematicians trained according to Western methods.[CS, P 142]

There were occasional small developments, and attempts to revive learning, but nothing of the magnitude of the previous millennium.

Worthy of a brief mention are both Kamalakara (c 1616-1700) and Jagannatha Samrat (c. 1690-1750). Both combined traditional ideas of Indian astronomy with Arabic (and some Greek) concepts, Kamalakara gave trigonometric results of interest and Samrat made several Sanskrit translations of Arabic ‘versions’ of Greek works, including notably Euclid’s Elements. However, as C Srinivasiengar comments:

…Jagannatha’s work was not mere translation.[CS, P 143]

Indeed his work contained new proofs not given by Euclid.

Under the patronage of monarch Sawai Jayasinha Raja (and his predecessor) Samrat was attempting, along with a group of scholars, to ‘reinvigorate’ science and learning in India. It must be admitted that these efforts do not appear to have been wholly successful, although the efforts were in the ‘greatest of faith’ and should be ‘applauded’. To some extent the work of these two scholars only serves to further highlight the lack of originality and indolent attitude that was present by this stage in the north of India, although their efforts should not be completely ignored.

It must be considered most unfortunate that a country, which on reflection was unarguably a world leader in the field of mathematics for several thousands of years, ceased to contribute in any significant way. A theory has been suggested that, had there been definite ‘links’ between each ‘major’ period we have discussed, then India would have led the world, unequivocally, in the field of mathematics and may have continued to for much longer. However discoveries that have been made in the last 150 years have significantly altered the chronology of Indian mathematics and the way in which we should view Indian contributions.

## 9: Keralese mathematics I. Introduction

The south western tip of India escaped the majority of the political upheaval, which engulfed the rest of the country, allowing a generally peaceful existence to continue. Thus the pursuit of scientific development was able to continue ‘uninterrupted’. It has only recently come to light that mathematics (and astronomy) continued to flourish in this area for several hundred years. Kerala mathematics was strongly influenced by astronomy, but this led to the derivation of mathematical results of huge importance. As a result of the recency of these discoveries it is quite probable that there are still further discoveries of ‘Kerala mathematics’ to be made, and a full analysis has yet to be carried out. However several findings have already been made that show several major concepts of renaissance European mathematics were first developed in India.

Indeed G Joseph quotes:

…In Kerala, the period between the1416^{th}and[GJ, P 287]^{th}centuries marked a high point in the indigenous development of (astronomy) and mathematics.

The works that have so far been analysed are of such a high level that it is though there may be missing links between the “classical period” and the medieval period of Kerala. There is also interest in the claims that European scholars may have had first hand knowledge of some Kerala mathematics, as the area was a focal point for trading with many parts of the world, including Europe. There is also some evidence of a transfer of technology between Europe and Kerala. I will discuss this issue in a little more depth later.

At this point in time only some interest is being paid to the recent discoveries that have been made, highlighting that in many historical ‘circles’ Indian developments are still not considered important. A further point of interest is that up to the 10^{th} century there was virtually no mathematical activity of note in the south of India. There are four areas in the south of the country – including Kerala in the south west corner, and with the exception of Mahavira (resident of Karnataka area) there was no mathematical output of significance until the 11^{th}-12^{th} century early Kerala literature. There are however impressions that Aryabhata I was a Keralite, and indeed, if this were true, then in the words of K Rajagopalan:

…Kerala would find a prominent place in the mathematical map of “our” country.[KR1, P 81]

Bhaskara I is also thought to have possibly been a Keralite.

Of the leading mathematicians of Kerala there is quite possibly more to be discovered but currently there are several whose work is of significant interest.

## 9 II. Mathematicians of Kerala

Narayana Pandit (c. 1340-1400), the earliest of the notable Keralese mathematicians, is known to have definitely written two works, an arithmetical treatise called *Ganita Kaumudi* and an algebraic treatise called *Bijganita Vatamsa*. He was strongly influenced by the work of Bhaskara II, which proves work from the classic period was known to Keralese mathematicians and was thus influential in the continued progress of the subject. Due to this influence Narayana is also thought to be the author of an elaborate commentary of Bhaskara II‘s *Lilavati*, titled *Karmapradipika* (or *Karma-Paddhati*). It has been suggested that this work was written in conjunction with another scholar, Sankara Variyar, while others attribute the work to Madhava (see later).

Although the *Karmapradipika* contains very little original work, seven different methods for squaring numbers are found within it, a contribution that is wholly original to the author. Narayana‘s other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, using the second order indeterminate equation *Nx*^{2} + 1 = *y*^{2}(Pell’s equation). Mathematical operations with zero, several geometrical rules and discussion of magic squares and similar figures are other contributions of note. Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II‘s work.

R Gupta has also brought to light Narayana‘s contributions to the topic of cyclic quadrilaterals. Subsequent developments of this topic, found in the works of Sankara Variyar and Ganesa interestingly show the influence of work of Brahmagupta.

Paramesvara (c. 1370-1460) is known to have been a pupil of Narayana Pandit, and also Madhava of Sangamagramma, who I will discuss later and is thought to have been a significant influence. He wrote commentaries on the work of Bhaskara I, Aryabhata I and Bhaskara II, and his contributions to mathematics include an outstanding version of the mean value theorem. Furthermore Paramesvara gave a mean value type formula for inverse interpolation of sine, and is thought to have been the first mathematician to give the radius of circle with inscribed cyclic quadrilateral, an expression that is normally attributed to Lhuilier (1782).

In turn, Nilakantha Somayaji (1444-1544) was a disciple of Paramesvara and was educated by his son Damodra. In his most notable work *Tantra Samgraha* (which ‘spawned’ a later anonymous commentary *Tantrasangraha-vyakhya* and a further commentary by the name *Yuktidipaika*, written in 1501) he elaborates and extends the contributions of Madhava. Sadly none of his mathematical works are extant, however it can be determined that he was a mathematician of some note. Nilakantha was also the author of *Aryabhatiya-bhasa* a commentary of the *Aryabhatiya*. Of great significance is the presence of *mathematical proof* (inductive) in Nilakantha‘s work.

Furthermore, his demonstration of particular cases of the series

tan

^{-1}t=t–t^{3}/3 +t^{5}/5 – … ,

when *t* = 1 and *t* = 1/3, and remarkably good rational approximations of p (using another Madhava series) are of great interest. Various results regarding infinite geometrically progressing convergent series are also attributed to Nilakantha

Citabhanu (1475-1550) has yet to find a place in books on Indian mathematics. His work on the solution of equations is quoted in a work called *Kriya-krama-kari*, by the scholar Sankara Variar, who is also relatively little known (although R Gupta mentions a further text, written by him).

Jyesthadeva (c. 1500-1575) was a member of the Kerala School, which was founded on the work of Madhava, Nilakantha, Paramesvara and others. His key work was the *Yukti-bhasa* (written in Malayalam, a regional language of Kerala). Similarly to the work of Nilakantha it is almost unique in the history of Indian mathematics, in that it contains both proofs of theorems and derivations of rules. He also studied various topics found in many previous Indian works, including integer solutions of systems of first degree equations solved using *kuttaka*.

## 9 III. Madhava of Sangamagramma

Although born in Cochin on the Keralese coast before the previous four scholars I have chosen to save my discussion of Madhava of Sangamagramma (c. 1340 – 1425) till last, as I consider him to be the greatest mathematician-astronomer of medieval India. Sadly all of his mathematical works are currently lost, although it is possible extant work may yet be ‘unearthed’. It is vaguely possible that he may have written *Karana Paddhati* a work written sometime between 1375 and 1475, but this is only speculative. All we know of Madhava comes from works of later scholars, primarily Nilakantha and Jyesthadeva. G Joseph also mentions surviving astronomical texts, but there is no mention of them in any other text I have consulted.

His most significant contribution was in moving on from the finite procedures of ancient mathematics to ‘treat their limit passage to infinity’, which is considered to be the essence of modern classical analysis. Although there is not complete certainty it is thought Madhava was responsible for the discovery of all of the following results:

1) = tan – (tan

^{3})/3 + (tan^{5})/5 – … , equivalent to Gregory series.2)

r= {r(rsin)/1(rcos)}-{r(rsin)^{3}/3(rcos)^{3}}+{r(rsin)^{5}/5(rcos)^{5}}- …3) sin = –

^{3}/3! +^{5}/5! – …, Madhava–Newton power series.4) cos = 1 –

^{2}/2! +^{4}/4! – …, Madhava-Newton power series.

Remembering that Indian sin = rsin, and Indian cos = rcos. Both the above results are occasionally attributed to Maclaurin.5)

p/4 1 – 1/3 + 1/5 – … 1/n(-f_{i}(n+1)),i= 1,2,3, and wheref_{1}=n/2,f_{2}= (n/2)/(n^{2}+ 1) andf_{3}= ((n/2)^{2}+ 1)/((n/2)(n^{2}+ 4 + 1))^{2}(a power series for p, attributed to Leibniz)6)

p/4 = 1 – 1/3 + 1/5 – 1/7 + … 1/n{-f(n+1)}, Euler‘s series.A particular case of the above series when

t=1/3 gives the expression:

7) p = 12 (1 – {1/(3 3)} + {1/(5 3^{2})} – {1/(7 3^{3})} + …}In generalisation of the expressions for f

_{2}and f_{3}as continued fractions, the scholar D Whiteside has shown that the correcting functionf(n) which makes ‘Euler’s’ series (of course it is not in fact Euler’s series) exact can be represented as an infinite continued fraction. There was no European parallel of this until W Brouncker‘s celebrated reworking in 1645 of J Wallis‘s related continued product.A further expression involving p:

8)pd2d+ 4d/(2^{2}– 1) – 4d/(4^{2}– 1) + … 4d/(n^{2}+ 1) etc, this resulted in improved approximations of p, a further term was added to the above expression, allowing Madhava to calculate p to 13 decimal places. The value p = 3.14159265359 is unique to Kerala and is not found in any other mathematical literature. A value correct to 17 decimal places (3.14155265358979324) is found in the workSadratnamala. R Gupta attributes calculation of this value to Madhava, (so perhaps he wrote this work, although this is pure conjecture).Of great interest is the following result:

9) tan^{-1}x=x–x^{3}/3 +x^{5}/5 – …, Madhava-Gregory series, power series for inverse tangent, still frequently attributed to Gregory and Leibniz.It is also expressed in the following way:

10) rarctan(y/x) =ry/x–ry^{3}/3x^{3}+ry^{5}/5x^{5}– …, wherey/x1The following results are also attributed to Madhava of Sangamagramma:

11) sin(x+h) sinx+ (h/r)cosx– (h^{2}/2r^{2})sinx12) cos(

x+h) cosx– (h/r)sinx– (h^{2}/2r^{2})cosxBoth the approximations for sine and cosine functions to the second order of small quantities, (see over page) are special cases of Taylor series, (which are attributed to B Taylor).

Finally, of significant interest is a further ‘Taylor’ series approximation of sine:

13) sin(x+h) sinx+ (h/r)cosx– (h^{2}/2r^{2})sinx+ (h^{3}/6r^{3})cosx.

Third order series approximation of the sine function usually attributed to Gregory.

With regards to this development R Gupta comments:

*…It is interesting that a four-term approximation formula for the sine function so close to the Taylor series approximation was known in India more than two centuries before the Taylor series expansion was discovered by Gregory about 1668.* [RG5, P 289]

Although these results all appear in later works, including the *Tantrasangraha* of Nilakantha and the *Yukti-bhasa* of Jyesthadeva it is generally accepted that all the above results originated from the work of Madhava. Several of the results are expressly attributed to him, for example Nilakantha quotes an alternate version of the sine series expansion as the work of Madhava. Further to these incredible contributions to mathematics, Madhava also extended some results found in earlier works, including those of Bhaskaracarya.

The work of Madhava is truly remarkable and hopefully in time full credit will be rewarded to his work, as C Rajagopal and M Rangachari note:

…Even if he be credited with only the discoveries of the series(sine and cosine expansions, see above,3)and4))at so unexpectedly early a date, assuredly merits a permanent place among the great mathematicians of the world.[CR /MR1, P 101]

Similarly G Joseph states:

…We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition.[GJ, P 293]

With regards to Keralese contributions as a whole, M Baron writes (in D Almeida, J John and A Zadorozhnyy):

…Some of the results achieved in connection with numerical integration by means of infinite series anticipate developments in Western Europe by several centuries. [DA/JJ/AZ1, P 79]

There remains a final Kerala work worthy of a brief mention, *Sadrhana-Mala* an astronomical treatise written by Sankara Varman serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19^{th} century and the author stands as the last notable name in Keralese mathematics.

In recent histories of mathematics there is acknowledgement that some of Madhava‘s remarkable results were indeed first discovered in India. This is clearly a positive step in redressing the imbalance but it seems unlikely that full ‘credit’ will be given for some time, as that will possibly require the re-naming of various series, which seems unlikely to happen!

Still in many quarters Keralese contributions go unnoticed, D Almeida, J John and A Zadorozhnyy note that a well known historian of mathematics makes:

…No acknowledgement of the work of the Keralese school.[DA/JJ/AZ1, P 78]

(Despite several Western publications of Keralese work.)

## 9 IV. Possible transmission of Keralese mathematics to Europe

In addition to my discussion, there is a very recent paper (written by D Almeida, J John and A Zadorozhnyy) of great interest, which goes as far as to suggest Keralese mathematics may have been transmitted to Europe. It is true that Kerala was in continuous contact with China, Arabia, and at the turn of the 16^{th} century, Europe, thus transmission might well have been possible. However the current theory is that Keralese calculus remained localised until its discovery by Charles Whish in the late 19^{th} century. There is no evidence of direct transmission by way of relevant manuscripts but there is evidence of methodological similarities, communication routes and a suitable chronology for transmission.

A key development of pre-calculus Europe, that of generalisation on the basis of *induction*, has deep methodological similarities with the corresponding Kerala development (200 years before). There is further evidence that John Wallis (1665) gave a recurrence relation and proof of Pythagoras theorem exactly as Bhaskara II did. The only way European scholars at this time could have been aware of the work of Bhaskara would have been through Keralese ‘routes’.

The need for greater calendar accuracy and inadequacies in sea navigation techniques are thought to have led Europeans to seek knowledge from their colonies throughout the 16^{th} and 17^{th}centuries. The requirements of calendar reform were imperative with the dating of Easter proving extremely problematic, by the 16^{th} century the European ‘Julian’ calendar was becoming so inaccurate that without correction Easter would eventually take place in summer! There were significant financial rewards for ‘anyone’ who could ‘assist’ in the improvement of navigation techniques. It is thought ‘information’ was sought from India in particular due to the influence of 11^{th} century Arabic translations of earlier Indian navigational methods.

Events also suggest it is quite possible that Jesuits (Christian missionaries) in Kerala were ‘encouraged’ to acquire mathematical knowledge while there.

It is feasible that these observations are mere coincidence but if indeed it is true that transmission of ideas and results between Europe and Kerala occurred, then the ‘role’ of later Indian mathematics is even more important than previously thought.

## 10: Conclusions

I wish to conclude initially by simply saying that the work of Indian mathematicians has been severely neglected by western historians, although the situation is improving somewhat. What I primarily wished to tackle was to answer two questions, firstly, why have Indian works been neglected, that is, what appears to have been the motivations and aims of scholars who have contributed to the Eurocentric view of mathematical history. This leads to the secondary question, why should this neglect be considered a great injustice.

I have attempted to answer this by providing a detailed investigation (and analysis) of many of the key contributions of the Indian subcontinent, and where possible, demonstrate how they pre-date European works (whether ancient Greek or later renaissance). I have further developed this ‘answer’ by providing significant evidence that a number of Indian works conversely influenced later European works, by way of Arabic transmissions. I have also included a discussion of the Indian decimal place value system which is undoubtedly the single greatest Indian contribution to the development of mathematics, and its wider applications in science, economics (and so on).

Discussing my first ‘question’ is less easy, as within the history of mathematics we find a variety of ‘stances’. If the most extreme Eurocentric model is ‘followed’ then all mathematics is considered European, and even less extreme stances do not give full credit to non-European contributions.

Indeed even in the very latest mathematics histories Indian ‘sections’ are still generally fairly brief. Why this attitude exists seems to be a cultural issue as much as anything. I feel it important not to be controversial or sweeping, but it is likely European scholars are resistant due to the way in which the inclusion of non-European, including Indian, contributions shakes up views that have been held for hundreds of years, and challenges the very foundations of the Eurocentric ideology. Perhaps what I am trying to say is that prior to discoveries made in technically fairly recent times, and in some cases actually recent times (say in the case of Kerala mathematics) it was generally believed that all science had been developed in Europe. It is almost more in the realms of psychology and culture that we argue about the effect the discoveries of non-European science may have had on the ‘psyche’ of European scholars.

However I believe this concept of ‘late discoveries’ is a relatively weak excuse, as there is substantial evidence that many European scholars were aware of some Indian works that had been translated into Latin. All that aside, there was significant resistance to scientific learning in its totality in Europe until at least the 14^{th}/15^{th} c and as a result, even though Spain is in Europe, there was little progression of Arabic mathematics throughout the rest of Europe during the Arab period.

However, *following* this period it seems likely Latin translations of Indian and Arabic works will have had an influence. It is possible that the scholars using them did not know the origin of these works. There has also been occasional evidence of European scholars taking results from Indian or Arabic works and presenting them as their own. Actions of this nature highlight the unscrupulous character of some European scholars.

Along with cultural reasons there are no doubt religious reasons for the neglect of Indian mathematics, indeed it was the power of the Christian church that contributed to the stagnation of learning, described as the dark ages, in Europe.

Above all, and regardless of the arguments, the simple fact is that many of the key results of mathematics, some of which are at the very ‘core’ of modern day mathematics, are of Indian origin. The results were almost all independently ‘rediscovered’ by European scholars during and after the ‘renaissance’ and while remarkable, history is something that should be complete and to neglect facts is both ignorant and arrogant. Indeed the neglect of Indian mathematical developments by many European scholars highlights what I can best describe as an idea of European “self importance”.

In many ways the results of the Indians were even more remarkable because they occurred so much earlier, that is, advanced mathematical ideas were developed by peoples considered less culturally and academically advanced than (late medieval) Europeans. Although this comment is controversial it may have been the motivation of several authors for neglect of Indian works, however, if this is the case, then opinions based on those attitudes should be ignored. Indian culture was of the highest standard, and this is reflected in the works that were produced.

Indian mathematicians made great strides in developing *arithmetic* (they can generally be credited with perfecting use of the operators), *algebra* (before Arab scholars), *geometry* (independent of the Greeks), and *infinite series expansions* and *calculus* (attributed to 17^{th}/18^{th} century European scholars). Also Indian works, through a variety of translations, have had significant influence throughout the world, from China, throughout the Arab Empire, and ultimately Europe.

To summarise, the main reasons for the neglect of Indian mathematics seem to be *religious, cultural* and *psychological*. Primarily it is because of an *ideological choice*. R Rashed mentions a concept of modernism vs. tradition. Furthermore Indian mathematics is criticised because it lacks rigour and is only interested in practical aims (which we know to be incorrect). Ultimately it is fundamentally important for historians to be neutral, (that includes Indian historians who may go too far the ‘other way’) and this has not always been the case, and indeed seems to still persist in some quarters.

In terms of consequences of the Eurocentric stance, it has undoubtedly resulted in a cultural divide and ‘angered’ non-Europeans scholars. There is an unhealthy air of European superiority, which is potentially quite politically dangerous, and scientifically unproductive. In order to maximise our knowledge of mathematics we must recognise many more nations as being able to provide valuable input, this statement is also relevant to past works. Eurocentrism has led to an historical ‘imbalance’, which basically means scholars are not presenting an accurate version of the history of the subject, which I view as unacceptable. Furthermore, it is vital to point out that European colonisation of India most certainly had an extremely negative effect on the progress of indigenous Indian science

At the very least it must be hoped that the history of Indian mathematics will, in time become as highly regarded, as I believe it should. As D Almeida, J John and A Zadorozhnyy comment:

…Awareness is not widespread.[DA/JJ/AZ1, P 78]

R Rashed meanwhile explains the current problem:

…The same representation is found time and again: classical science, both in modernity and historicity appears in the final count as work of European humanityalone…

He continues:

*…It is true that the existence of some scientific activity in other cultures is occasionally acknowledged. Nevertheless, it remains outside history or is only integrated in so far as it contributed to science, which is essentially European.* [RR, P 333]

In short, the doctrine of the western essence of classical science does not take objective history into account.

Finally, beyond simply alerting people to the remarkable developments of Indian mathematicians between around 3000 BC and 1600 AD, and challenging the Eurocentric ideology of the history of the subject, it is thought further analysis and research could also have important consequences for future developments of the subject.

It is thought analysis of the difference in the epistemologies of 17^{th} century European and 15^{th} century Keralese calculus could help to provide an answer to the controversial issue of whether mathematics should concern itself with proof or calculation. Furthermore, in terms of the way mathematics is currently ‘taught’ D Almeida, J John and A Zadorozhnyy elucidate:

…The floating point numbers were used by Kerala mathematicians and, using this system of numbers, they were able to investigate and rationalise about the convergence of series. So we(DA/JJ/AZ)believe that a study of Keralese calculus will provide insights into computer-assisted teaching strategies.[DA/JJ/AZ, P 96]

(N.B. computers use a floating-point number system.)

Clearly there is massive scope for further study in the area of the history of Indian and other non-European mathematics, and it is still a topic on which relatively few works have been written, although slowly significantly more attention is being paid to the contributions of non-European countries.

In specific reference to my own project, I would have liked to have been able to go into more depth in my discussion of Indian algebra, and given many more worked examples, as I consider Indian algebra to be both remarkable and severely neglected. Furthermore there is scope for significant and important study of the transmission of Indian mathematics across the world, especially into Europe, via Arabic and later Keralese routes. It is clear that there are many more discoveries to be made and much more that can be written, as C Srinivasiengar observes:

…The last word on the history of ancient civilisation will never be said.[CS, P 1]

As a final note, many question the worth of historical study, beyond personal interest, but I hope I have shown in the course of my work some of the value and importance of historical study. I will conclude with a quote from the scholar G Miller, who commented:

…The history of mathematics is the only one of the sciences to possess a considerable body of perfect and inspiring results which were proved 2000 years ago by the same thought processes as are used today. This history is therefore useful for directing attention to the permanent value of scientific achievements and the great intellectual heritage, which these achievements present, to the world.[AA’D, P 11]