The Indian Sulbasutras

The Vedic people entered India about 1500 BC from the region that today is Iran. The word Vedic describes the religion of these people and the name comes from their collections of sacred texts known as the Vedas. The texts date from about the 15^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 SX² = PX² + PS².

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.

EQ² = QR² - RE²
= QP² - YP²
= 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 [Indian J. Hist. Sci. 13 (1) (1978), 32-41.',6)">6]. In [Centaurus 31 (2) (1988), 114-125.',3)">3] the value π = ²⁵/₈ = 3.125 is found in the Manava Sulbasutras.

In [A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin, Studies in the history of mathematics 19 \'Nauka\' (Moscow, 1974), 220-222; 302.',9)">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.
Consider the diagram on the right.

The Sulbasutras also examine the converse problem of finding a circle equal in area to a given square. The following construction appears. Given a square ABCD find the centre O. Rotate OD to position OE where E is the midpoint 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 2a then the radius of the circle is r where

r = OE - EQ
= √2a - ²/₃(√2a - a)
= a (^√2/₃ + ²/₃).

Then 4a ² = πa² (^√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

√2 = 1 + 1/3 + 1/(3 4) - 1/(3 4 34) = 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 [The science of the Sulba (Calcutta, 1932).',1)">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

1 + 1/3 + 1/(3 4)

which, of course, is the first three terms of the approximation. Now Datta argues in [The science of the Sulba (Calcutta, 1932).',1)">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

2 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

1 + 1/3 + 1/(3 4) - 1/(3 4 34)

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

2 x (1 + ¹/₃ + ¹/₁₂) - x² = (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 [Math. Education 6 (1972), B77-B79.',4)">4] Gupta gives a simpler way of obtaining the approximation for √2 than that given by Datta in [The science of the Sulba (Calcutta, 1932).',1)">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.

References (9 books/articles)

Indian numerals

It is worth beginning this article with the same quote from Laplace which we give in the article Overview of Indian mathematics. Laplace wrote:-

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 [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)">1] Ifrah lists a number of the hypotheses which have been put forward.

1. The Brahmi numerals came from the Indus valley culture of around 2000 BC.

2. The Brahmi numerals came from Aramaean numerals.

3. The Brahmi numerals came from the Karoshthi alphabet.

4. The Brahmi numerals came from the Brahmi alphabet.

5. The Brahmi numerals came from an earlier alphabetic numeral system, possibly due to Panini.

6. 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):

I, II, III, X, IX, IIX, IIIX, XX.

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 [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)">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 [Indian J. Hist. Sci. 18 (1) (1983), 23-38.',7)">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:

1. a donation charter of Dhiniki dated 794 in the Vikrama calendar which translates to a date in our calendar of 737 AD.

2. an inscription of Devendravarman dated 675 in the Shaka calendar which translates to a date in our calendar of 753 AD.

3. a donation charter of Danidurga dated 675 in the Shaka calendar which translates to a date in our calendar of 737 AD.

4. a donation charter of Shankaragana dated 715 in the Shaka calendar which translates to a date in our calendar of 793 AD.

5. a donation charter of Nagbhata dated 872 in the Vikrama calendar which translates to a date in our calendar of 815 AD.

6. 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 [Arch. Hist. Exact Sci. 37 (4) (1987), 365-392.',8)">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 [The crest of the peacock (London, 1991).',2)">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 10⁷. He lists the powers of 10 up to 10⁵³. Taking this as a first level he then carries on to a second level and gets eventually to 10⁴²¹. 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 [The crest of the peacock (London, 1991).',2)">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 [The nothing that is : a natural history of zero (London, 1999).',3)">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.

References (11 books/articles)

An overview of Indian mathematics

It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them.

We shall examine the contributions of Indian mathematics in this article, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested. Laplace put this with great clarity:-

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 the 3rd 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 [Ganita 19 (2) (1968), 49-72.',25)">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 [Math. Student 53 (1-4) (1985), 204-208',26)">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 [Math. Student 53 (1-4) (1985), 204-208',26)">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

π R = 4R - 4R/3 + 4R/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 [Math. Student 13 (1945), 92-98.',15)">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, x² + y² = d, x² - y² = e, x³ + y³ = f, and x³ - y³ = 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 [Historia Math. 25 (1) (1998), 1-21.',12)">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.

References (28 books/articles)

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