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A STUDY ON ANCIENT INDIAN MATHEMATICS

A Study on Ancient 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..

1

2

3

4

5

6

7

8

9

Ξ

+

h

BRAHMI NUMERALS AROUND 1ST CENTURY A.D.

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 6th 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.

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:-

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 zheory of quadratic equations and the method of successive approximations to solving problems in spherical zstronomy.

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:-

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. 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 [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, xy = b, xy = c, x2 + y2 = d, x2y2 = e, x3 + y3 = f, and x3y3 = 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.

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

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 quot;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 15th 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.

1

2

3

Ξ

BRAHMI
NUMERALS 1, 2 , 3

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 4th century AD. Of course different inscriptions differ somewhat in the style of the symbols.

Here is One Style of the Brahmi Numerals.

1

2

3

4

5

6

7

8

9

Ξ

+

h

BRAHMI NUMERALS AROUND 1ST CENTURY A.D.

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. 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, 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:

1

2

3

4

5

6

7

8

9

|

||

|||

|| ||

||| ||

||| |||

|||| |||

|||| ||||

||| ||| |||

IFRAH’S
GUESS FOR PRE-BRAHMI NUMBERS

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.

1

2

3

4

5

6

7

8

9

=

Ξ

؏

ף

GUPTA
NUMERALS AROUND 4TH CENTURY A.D.

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 4th century AD to the late 6th century AD.
The Gupta numerals developed from the Brahmi numerals and were spread over
large areas by the Gupta empire as they conquered territory.

1

2

3

4

5

6

7

8

9

0

؏

NAGARI
NUMERALS AROUND 11TH CENTURY A.D.

The Gupta numerals evolved into the Nagari numerals, sometimes called the Devanagari numerals. This form evolved from the Gupta numerals beginning around the 7th century AD and continued to develop from the 11th 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 7th to the 16th centuries in examined in detail. In this paper, however, Gupta claims that Indian numerals had reached Southern Europe by the end of the 5th 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 11th 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 19th 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 6th 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. 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 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:-

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 importanceon the development of mathematics.

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 15th to the 5th 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 KATYAYANA FORM OF PYTHAGORAS’S THEOREM

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 quares.

Consider the diagram on the right.

TO FIND A SQUARE EQUAL TO THE SUM OF THE TWO GIVEN SQUARES – GIVEN ALL SULBASUTRAS

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
= QP2YP2
= 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 cross(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. The value π = 25/8 = 3.125 is found in the Manava Sulbasutras.

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 = OEEQ
= √2a2/3(√2aa)
= 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 cross4)
– 1/(3 cross4 cross34)
= 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.

DATTA’S SUGGESTION OF HOW APASTAMBA FOUND HIS APPRXIMATION OF √2

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 cross4)

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

2 crossx
cross(1 + 1/3 + 1/12) = (1/12)2.

This has the solution x = 1/(3 cross4 cross34)
which is approximately 0.002450980392. We now have a square the length of whose sides is

1 + 1/3 + 1/(3 cross4)
– 1/(3 cross4 cross34)

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 crossx
cross(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!

Gupta gives a simpler way of obtaining the approximation for √2 than that given by Datta. 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

It is a
little hard to define Jaina mathematics. Jainism is a religion and philosophy
which was founded in India around the 6th 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 5th century BC up to the 18th 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 4th 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 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 2588 years. Note that 2588 is
a very large number!

2588
= 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 rq
(taken to be the radius of the earth) and having a fixed height h. The
number nq = f(rq) is the number of
very tiny white mustard seeds that can be placed in this container. Next, r1

= g(rq) is defined by a complicated recursive
subprocedure, and then as before a new larger number n1 = f(r1)
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 296.

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 an early mathematical manuscript which was discovered
over 100 years ago. We shall discuss in a moment the problem of dating this
manuscript, a topic which has aroused much controversy, but for the moment we
will examine how it was discovered. The paper [8] describes this discovery
along with the early history of the manuscript. Gupta writes:-

The
Bakhshali Manuscript is the name given to the mathematical work written on
birch bark and found in the summer of
1881 near the village Bakhshali (or
Bakhshalai
) of the Yusufzai subdivision of the Peshawar district (now
in Pakistan
). The village is in Mardan tahsil and is situated 50 miles
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 in
1882, and this was published in the Indian
Antiquary in
1883. He gave a fuller account at the Seventh Oriental
Conference held at Vienna in
1886 and this was published in its Proceedings.
A revised version of this paper appeared in the Indian Antiquary of
1888.
In
1902, 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 5th 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 3/41/2 would be written.

3/4
minus ½

 

 

3

1+

MEANS ¾
MINUS 1/2

4

2

 

 

 

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

1 plus 1/3

 

 

1

1

3

MEANS 1
PLUS 1/3 SO EQUALS TO 4/3

 

 

 

and 1 minus 1/3
= 2/3 in the following way

1 minus 1/3
= 2/3

 

 

1

1

3+

MEANS 1
MINUS 1/3 SO EQUALS TO 2/3

 

 

 

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

 

 

5

2

YU

PHA 7

1

1

THIS MEANS 5/1 PLUS 2/1 EQUALS TO
7

 

5/1
plus 2/1

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

 

 

 

 

1

1

3+

BHA 8

PHA 12

THIS MEANS 8 DIVIDED BY 2/3 EQUALS
TO 12

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.

 

 

 

 

1

1

1

BHA 32

PHALA 108

1

1

1

1

 

3+

3+

3+

 

 

FIND NUMBER 32 DIVIDED BY (2/3)2.
IT EQUALS TO 108

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 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 x1, x2, x3 and k

(where x1 is the price of an asava, x2 is
the price of a haya, and x3 is the price of a horse)
satisfying

5 x1

+ x2 + x3 = x1 + 7 x2+
x3 = x1 + x2 + 8 x3

= k.

Then 4 x1
= 6 x2 = 7 x3 = k – (x1

+ x2 + x3).

For integer
solutions k – (x1 + x2 + x3)
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 – (x1 + x2

+ x3) = 168 (this is 4 cross6 cross7)
giving x1 = 42, x2 = 28, x3

= 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 x1,
x2, x3 in terms of k to obtain

x1 = 21k/131,
x2 = 14k/131, x3 = 12k/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 – (x1
+ x2 + x3) = 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 gets
13/6
dinaras a day and the other
3/2. The first owes
the second
10 dinaras. calculate and tell me when they have equal
amounts.

Now I would
solve this by saying that the first gets 13/63/2
= 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 cross30/6

– 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

8 50 12

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

phala crossiccha/pramana

or in the
case of the example 50 cross12/8
= 75 dinaras.

Applying
this to the page-boy problem we obtain equal amounts for the page-boys after n

days where

13 crossn/6
= 3 crossn/2
+20

so n

= 30 and each has 13 cross30/6
– 10 = 55 dinaras.

Another
interesting piece of mathematics in the manuscript concerns calculating square
roots. The following formula is used

Q
= √(A2 + b) = A + b/2A – (b/2A)2/(2(A

+ b/2A))

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:

√487
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 = 302
– 11 instead of 292 + 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.

‘Zero’
Origin

One of the
commonest questions which the readers of this archive ask is: Who discovered
zero? Why then have we not written an article on zero as one of the first in
the archive? The reason is basically because of the difficulty of answering the
question in a satisfactory form. If someone had come up with the concept of
zero which everyone then saw as a brilliant innovation to enter mathematics
from that time on, the question would have a satisfactory answer even if we did
not know which genius invented it. The historical record, however, shows quite
a different path towards the concept. Zero makes shadowy appearances only to
vanish again almost as if mathematicians were searching for it yet did not
recognise its fundamental significance even when they saw it.

The first
thing to say about zero is that there are two uses of zero which are both
extremely important but are somewhat different. One use is as an empty place
indicator in our place-value number system. Hence in a number like 2106 the
zero is used so that the positions of the 2 and 1 are correct. Clearly 216
means something quite different. The second use of zero is as a number itself
in the form we use it as 0. There are also different aspects of zero within
these two uses, namely the concept, the notation, and the name. (Our name
“zero” derives ultimately from the Arabic sifr which also
gives us the word "cipher".)

Neither of
the above uses has an easily described history. It just did not happen that
someone invented the ideas, and then everyone started to use them. Also it is
fair to say that the number zero is far from an intuitive concept. Mathematical
problems started as ‘real’ problems rather than abstract problems. Numbers in
early historical times were thought of much more concretely than the abstract
concepts which are our numbers today. There are giant mental leaps from 5
horses to 5 “things” and then to the abstract idea of
“five”. If ancient peoples solved a problem about how many horses a
farmer needed then the problem was not going to have 0 or -23 as an answer.

One might
think that once a place-value number system came into existence then the 0 as
an empty place indicator is a necessary idea, yet the Babylonians had a
place-value number system without this feature for over 1000 years. Moreover
there is absolutely no evidence that the Babylonians felt that there was any
problem with the ambiguity which existed. Remarkably, original texts survive
from the era of Babylonian mathematics. The Babylonians wrote on tablets of
unbaked clay, using cuneiform writing. The symbols were pressed into soft clay
tablets with the slanted edge of a stylus and so had a wedge-shaped appearance
(and hence the name cuneiform). Many tablets from around 1700 BC survive and we
can read the original texts. Of course their notation for numbers was quite
different from ours (and not based on 10 but on 60) but to translate into our
notation they would not distinguish between 2106 and 216 (the context would
have to show which was intended). It was not until around 400 BC that the
Babylonians put two wedge symbols into the place where we would put zero to
indicate which was meant, 216 or 21 ” 6.

The two
wedges were not the only notation used, however, and on a tablet found at Kish,
an ancient Mesopotamian city located east of Babylon in what is today
south-central Iraq, a different notation is used. This tablet, thought to date
from around 700 BC, uses three hooks to denote an empty place in the positional
notation. Other tablets dated from around the same time use a single hook for
an empty place. There is one common feature to this use of different marks to
denote an empty position. This is the fact that it never occured at the end of
the digits but always between two digits. So although we find 21 ” 6 we never
find 216 ”. One has to assume that the older feeling that the context was
sufficient to indicate which was intended still applied in these cases.

If this
reference to context appears silly then it is worth noting that we still use
context to interpret numbers today. If I take a bus to a nearby town and ask
what the fare is then I know that the answer “It’s three fifty” means
three pounds fifty pence. Yet if the same answer is given to the question about
the cost of a flight from Edinburgh to New York then I know that three hundred
and fifty pounds is what is intended.

We can see
from this that the early use of zero to denote an empty place is not really the
use of zero as a number at all, merely the use of some type of punctuation mark
so that the numbers had the correct interpretation.

Now the
ancient Greeks began their contributions to mathematics around the time that
zero as an empty place indicator was coming into use in Babylonian mathematics.
The Greeks however did not adopt a positional number system. It is worth
thinking just how significant this fact is. How could the brilliant
mathematical advances of the Greeks not see them adopt a number system with all
the advantages that the Babylonian place-value system possessed? The real answer
to this question is more subtle than the simple answer that we are about to
give, but basically the Greek mathematical achievements were based on geometry.
Although Euclid’s Elements contains a book on number theory, it is based
on geometry. In other words Greek mathematicians did not need to name their
numbers since they worked with numbers as lengths of lines. Numbers which
required to be named for records were used by merchants, not mathematicians,
and hence no clever notation was needed.

Now there were
exceptions to what we have just stated. The exceptions were the mathematicians
who were involved in recording astronomical data. Here we find the first use of
the symbol which we recognise today as the notation for zero, for Greek
astronomers began to use the symbol O. There are many theories why this
particular notation was used. Some historians favour the explanation that it is
omicron, the first letter of the Greek word for nothing namely
“ouden”. Neugebauer, however, dismisses this explanation since the
Greeks already used omicron as a number – it represented 70 (the Greek number
system was based on their alphabet). Other explanations offered include the
fact that it stands for “obol”, a coin of almost no value, and that
it arises when counters were used for counting on a sand board. The suggestion
here is that when a counter was removed to leave an empty column it left a
depression in the sand which looked like O.

Ptolemy in
the Almagest written around 130 AD uses the Babylonian sexagesimal
system together with the empty place holder O. By this time Ptolemy is using
the symbol both between digits and at the end of a number and one might be
tempted to believe that at least zero as an empty place holder had firmly
arrived. This, however, is far from what happened. Only a few exceptional
astronomers used the notation and it would fall out of use several more times
before finally establishing itself. The idea of the zero place (certainly not
thought of as a number by Ptolemy who still considered it as a sort of
punctuation mark) makes its next appearance in Indian mathematics.

The scene
now moves to India where it is fair to say the numerals and number system was
born which have evolved into the highly sophisticated ones we use today. Of
course that is not to say that the Indian system did not owe something to
earlier systems and many historians of mathematics believe that the Indian use
of zero evolved from its use by Greek astronomers. As well as some historians
who seem to want to play down the contribution of the Indians in a most
unreasonable way, there are also those who make claims about the Indian
invention of zero which seem to go far too far. For example Mukherjee in [6]
claims:-

… the
mathematical conception of zero … was also present in the spiritual form from
17 000 years back in India.

What is
certain is that by around 650AD the use of zero as a number came into Indian
mathematics. The Indians also used a place-value system and zero was used to
denote an empty place. In fact there is evidence of an empty place holder in
positional numbers from as early as 200AD in India but some historians dismiss
these as later forgeries. Let us examine this latter use first since it
continues the development described above.

In around
500AD Aryabhata devised a number system which has no zero yet was a positional
system. He used the word “kha” for position and it would be used
later as the name for zero. There is evidence that a dot had been used in
earlier Indian manuscripts to denote an empty place in positional notation. It
is interesting that the same documents sometimes also used a dot to denote an
unknown where we might use x. Later Indian mathematicians had names for
zero in positional numbers yet had no symbol for it. The first record of the
Indian use of zero which is dated and agreed by all to be genuine was written
in 876.

We have an
inscription on a stone tablet which contains a date which translates to 876.
The inscription concerns the town of Gwalior, 400 km south of Delhi, where they
planted a garden 187 by 270 hastas which would produce enough flowers to allow
50 garlands per day to be given to the local temple. Both of the numbers 270
and 50 are denoted almost as they appear today although the 0 is smaller and
slightly raised.

We now come to
considering the first appearance of zero as a number. Let us first note that it
is not in any sense a natural candidate for a number. From early times numbers
are words which refer to collections of objects. Certainly the idea of number
became more and more abstract and this abstraction then makes possible the
consideration of zero and negative numbers which do not arise as properties of
collections of objects. Of course the problem which arises when one tries to
consider zero and negatives as numbers is how they interact in regard to the
operations of arithmetic, addition, subtraction, multiplication and division.
In three important books the Indian mathematicians Brahmagupta, Mahavira and
Bhaskara tried to answer these questions.

Brahmagupta
attempted to give the rules for arithmetic involving zero and negative numbers
in the seventh century. He explained that given a number then if you subtract
it from itself you obtain zero. He gave the following rules for addition which
involve zero:-

The sum of
zero and a negative number is negative, the sum of a positive number and zero
is positive, the sum of zero and zero is zero.

Subtraction
is a little harder:-

A negative
number subtracted from zero is positive, a positive number subtracted from zero
is negative, zero subtracted from a negative number is negative, zero
subtracted from a positive number is positive, zero subtracted from zero is
zero.

Brahmagupta
then says that any number when multiplied by zero is zero but struggles when it
comes to division:-

A positive
or negative number when divided by zero is a fraction with the zero as
denominator. Zero divided by a negative or positive number is either zero or is
expressed as a fraction with zero as numerator and the finite quantity as
denominator. Zero divided by zero is zero.

Really
Brahmagupta is saying very little when he suggests that n divided by
zero is n/0. Clearly he is struggling here. He is certainly wrong when
he then claims that zero divided by zero is zero. However it is a brilliant
attempt from the first person that we know who tried to extend arithmetic to
negative numbers and zero.

In 830,
around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita
Sara Samgraha
which was designed as an updating of Brahmagupta’s book. He
correctly states that:-

… a number
multiplied by zero is zero, and a number remains the same when zero is
subtracted from it.

However his
attempts to improve on Brahmagupta’s statements on dividing by zero seem to
lead him into error. He writes:-

A number
remains unchanged when divided by zero.

Since this
is clearly incorrect my use of the words "seem to lead him into
error" might be seen as confusing. The reason for this phrase is that some
commentators on Mahavira have tried to find excuses for his incorrect
statement.

Bhaskara
wrote over 500 years after Brahmagupta. Despite the passage of time he is still
struggling to explain division by zero. He writes:-

A quantity
divided by zero becomes a fraction the denominator of which is zero. This fraction
is termed an infinite quantity. In this quantity consisting of that which has
zero for its divisor, there is no alteration, though many may be inserted or
extracted; as no change takes place in the infinite and immutable God when
worlds are created or destroyed, though numerous orders of beings are absorbed
or put forth.

So Bhaskara
tried to solve the problem by writing n/0 = ∞. At first sight we
might be tempted to believe that Bhaskara has it correct, but of course he does
not. If this were true then 0 times ∞ must be equal to every number n,
so all numbers are equal. The Indian mathematicians could not bring themselves to
the point of admitting that one could not divide by zero. Bhaskara did
correctly state other properties of zero, however, such as 02 = 0,
and √0 = 0.

Perhaps we
should note at this point that there was another civilisation which developed a
place-value number system with a zero. This was the Maya people who lived in
central America, occupying the area which today is southern Mexico, Guatemala,
and northern Belize. This was an old civilisation but flourished particularly
between 250 and 900. We know that by 665 they used a place-value number system
to base 20 with a symbol for zero. However their use of zero goes back further
than this and was in use before they introduced the place-valued number system.
This is a remarkable achievement but sadly did not influence other peoples.

The
brilliant work of the Indian mathematicians was transmitted to the Islamic and
Arabic mathematicians further west. It came at an early stage for al-Khwarizmi
wrote Al’Khwarizmi on the Hindu Art of Reckoning which describes the
Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and
0. This work was the first in what is now Iraq to use zero as a place holder in
positional base notation. Ibn Ezra, in the 12th century, wrote three
treatises on numbers which helped to bring the Indian symbols and ideas of
decimal fractions to the attention of some of the learned people in Europe. The
Book of the Number
describes the decimal system for integers with place
values from left to right. In this work ibn Ezra uses zero which he calls
galgal (meaning wheel or circle). Slightly later in the 12th century
al-Samawal was writing:-

If we
subtract a positive number from zero the same negative number remains. … if
we subtract a negative number from zero the same positive number remains.

The Indian
ideas spread east to China as well as west to the Islamic countries. In 1247
the Chinese mathematician Ch’in Chiu-Shao wrote Mathematical treatise in
nine sections
which uses the symbol O for zero. A little later, in 1303,
Zhu Shijie wrote Jade mirror of the four elements which again uses the
symbol O for zero.

Fibonacci
was one of the main people to bring these new ideas about the number system to
Europe. As the authors of [12] write:-

An important
link between the Hindu-Arabic number system and the European mathematics is the
Italian mathematician Fibonacci.

In Liber
Abaci
he described the nine Indian symbols together with the sign 0 for
Europeans in around 1200 but it was not widely used for a long time after that.
It is significant that Fibonacci is not bold enough to treat 0 in the same way
as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the
“sign” zero while the other symbols he speaks of as numbers. Although
clearly bringing the Indian numerals to Europe was of major importance we can
see that in his treatment of zero he did not reach the sophistication of the
Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic
mathematicians such as al-Samawal.

One might
have thought that the progress of the number systems in general, and zero in
particular, would have been steady from this time on. However, this was far
from the case. Cardan solved cubic and quartic equations without using zero. He
would have found his work in the 1500’s so much easier if he had had a zero but
it was not part of his mathematics. By the 1600’s zero began to come into
widespread use but still only after encountering a lot of resistance.

Of course
there are still signs of the problems caused by zero. Recently many people
throughout the world celebrated the new millennium on 1 January 2000. Of course
they celebrated the passing of only 1999 years since when the calendar was set
up no year zero was specified. Although one might forgive the original error,
it is a little surprising that most people seemed unable to understand why the
third millennium and the 21st century begin on 1 January 2001. Zero
is still causing problems!

Pi (∏)
Origin

A little
known verse of the Bible reads

And he made
a molten sea, ten cubits from the one brim to the other: it was round all
about, and his height was five cubits: and a line of thirty cubits did compass
it about.
(I Kings 7, 23)

The same
verse can be found in II Chronicles 4, 2. It occurs in a list of specifications
for the great temple of Solomon, built around 950 BC and its interest here is
that it gives π = 3. Not a very accurate value of course and not even very
accurate in its day, for the Egyptian and Mesopotamian values of 25/8

= 3.125 and √10 = 3.162 have been traced to much earlier dates: though in
defence of Solomon’s craftsmen it should be noted that the item being described
seems to have been a very large brass casting, where a high degree of
geometrical precision is neither possible nor necessary. There are some
interpretations of this which lead to a much better value.

The fact
that the ratio of the circumference to the diameter of a circle is constant has
been known for so long that it is quite untraceable. The earliest values of
π including the ‘Biblical’ value of 3, were almost certainly found by
measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there
is good evidence for 4 cross(8/9)2
= 3.16 as a value for π.

The first
theoretical calculation seems to have been carried out by Archimedes of
Syracuse (287-212 BC). He obtained the approximation

223/71
< π < 22/7.

Before
giving an indication of his proof, notice that very considerable sophistication
involved in the use of inequalities here. Archimedes knew, what so many people
to this day do not, that π does not equal 22/7, and
made no claim to have discovered the exact value. If we take his best estimate
as the average of his two bounds we obtain 3.1418, an error of about 0.0002.

Here is
Archimedes’ argument.

Consider a
circle of radius 1, in which we inscribe a regular polygon of 3 cross2n-1

sides, with semiperimeter bn, and superscribe a regular
polygon of 3 cross2n-1
sides, with semiperimeter an.

The diagram
for the case n = 2 is on the right.

The effect
of this procedure is to define an increasing sequence

b1 , b2
, b3 , …

and a
decreasing sequence

a1 , a2
, a3 , …

such that
both sequences have limit π.

Using
trigonometrical notation, we see that the two semiperimeters are given by

an = K
tan(π/K), bn = K sin(π/K),

where K
= 3 cross2n-1.
Equally, we have

an+1 = 2K

tan(π/2K), bn+1 = 2K
sin(π/2K),

and it is
not a difficult exercise in trigonometry to show that

(1/an
+ 1/bn) = 2/an+1   .
. . (1)

an+1bn = (bn+1)2       .
. . (2)

Archimedes,
starting from a1 = 3 tan(π/3) = 3√3 and b1

= 3 sin(π/3) = 3√3/2, calculated a2 using (1),
then b2 using (2), then a3 using (1), then b3

using (2), and so on until he had calculated a6 and b6.
His conclusion was that

b6 < π
< a6 .

It is
important to realise that the use of trigonometry here is unhistorical:
Archimedes did not have the advantage of an algebraic and trigonometrical
notation and had to derive (1) and (2) by purely geometrical means. Moreover he
did not even have the advantage of our decimal notation for numbers, so that
the calculation of a6 and b6 from (1) and
(2) was by no means a trivial task. So it was a pretty stupendous feat both of
imagination and of calculation and the wonder is not that he stopped with
polygons of 96 sides, but that he went so far.

For of
course there is no reason in principle why one should not go on. Various people
did, including:

Ptolemy

(c. 150
AD)

3.1416

Zu
Chongzhi

(430-501
AD)

355/113

al-Khwarizmi

(c. 800 )

3.1416

al-Kashi

(c. 1430)

14 places

Viète

(1540-1603)

9 places

Roomen

(1561-1615)

17 places

Van Ceulen

(c. 1600)

35 places

Except for
Zu Chongzhi, about whom next to nothing is known and who is very unlikely to
have known about Archimedes’ work, there was no theoretical progress involved
in these improvements, only greater stamina in calculation. Notice how the
lead, in this as in all scientific matters, passed from Europe to the East for
the millennium 400 to 1400 AD.

Al-Khwarizmi
lived in Baghdad, and incidentally gave his name to ‘algorithm’, while the
words al jabr in the title of one of his books gave us the word
‘algebra’. Al-Kashi lived still further east, in Samarkand, while Zu Chongzhi,
one need hardly add, lived in China.

The European
Renaissance brought about in due course a whole new mathematical world. Among
the first effects of this reawakening was the emergence of mathematical
formulae for π. One of the earliest was that of Wallis (1616-1703)

2/π =
(1.3.3.5.5.7. …)/(2.2.4.4.6.6. …)

and one of
the best-known is

π/4
= 1 – 1/3 + 1/51/7

+ ….

This formula
is sometimes attributed to Leibniz (1646-1716) but is seems to have been first
discovered by James Gregory (1638- 1675).

These are
both dramatic and astonishing formulae, for the expressions on the right are
completely arithmetical in character, while π arises in the first instance
from geometry. They show the surprising results that infinite processes can
achieve and point the way to the wonderful richness of modern mathematics.

From the
point of view of the calculation of π, however, neither is of any use at
all. In Gregory’s series, for example, to get 4 decimal places correct we
require the error to be less than 0.00005 = 1/20000, and
so we need about 10000 terms of the series. However, Gregory also showed the
more general result

tan-1
x = xx3/3 + x5/5 – … (-1
x ≤ 1)   . . . (3)

from which
the first series results if we put x = 1. So using the fact that

tan-1(1/√3)
= π/6 we get

π/6
= (1/√3)(1 – 1/(3.3) + 1/(5.3.3) – 1/(7.3.3.3) +

which
converges much more quickly. The 10th term is 1/(19 cross39√3),
which is less than 0.00005, and so we have at least 4 places correct after just
9 terms.

An even
better idea is to take the formula

π/4
= tan-1(1/2) + tan-1(1/3)   .
. . (4)

and then
calculate the two series obtained by putting first 1/2
and the 1/3 into (3).

Clearly we
shall get very rapid convergence indeed if we can find a formula something like

π/4
= tan-1(1/a) + tan-1(1/b)

with a
and b large. In 1706 Machin found such a formula:

π/4
= 4 tan-1(1/5) – tan-1(1/239)   .
. . (5)

Actually
this is not at all hard to prove, if you know how to prove (4) then there is no
real extra difficulty about (5), except that the arithmetic is worse. Thinking
it up in the first place is, of course, quite another matter.

With a
formula like this available the only difficulty in computing π is the
sheer boredom of continuing the calculation. Needless to say, a few people were
silly enough to devote vast amounts of time and effort to this tedious and
wholly useless pursuit. One of them, an Englishman named Shanks, used Machin’s
formula to calculate π to 707 places, publishing the results of many years
of labour in 1873. Shanks has achieved immortality for a very curious reason
which we shall explain in a moment.

Here is a
summary of how the improvement went:

1699:

Sharp used
Gregory’s result to get 71 correct digits

1701:

Machin
used an improvement to get 100 digits and the following used his methods:

1719:

de Lagny
found 112 correct digits

1789:

Vega got
126 places and in 1794 got 136

1841:

Rutherford
calculated 152 digits and in 1853 got 440

1873:

Shanks
calculated 707 places of which 527 were correct

A more
detailed Chronology is available.

Shanks knew
that π was irrational since this had been proved in 1761 by Lambert.
Shortly after Shanks’ calculation it was shown by Lindemann that π is
transcendental, that is, π is not the solution of any polynomial equation
with integer coefficients. In fact this result of Lindemann showed that
‘squaring the circle’ is impossible. The transcendentality of π implies
that there is no ruler and compass construction to construct a square equal in
area to a given circle.

Very soon
after Shanks’ calculation a curious statistical freak was noticed by De Morgan,
who found that in the last of 707 digits there was a suspicious shortage of
7’s. He mentions this in his Budget of Paradoxes of 1872 and a curiosity
it remained until 1945 when Ferguson discovered that Shanks had made an error
in the 528th place, after which all his digits were wrong. In 1949 a
computer was used to calculate π to 2000 places. In this and all
subsequent computer expansions the number of 7’s does not differ significantly
from its expectation, and indeed the sequence of digits has so far passed all
statistical tests for randomness.

We should
say a little of how the notation π arose. Oughtred in 1647 used the symbol
d/π for the ratio of the diameter of a circle to its circumference.
David Gregory (1697) used π/r for the ratio of the circumference of
a circle to its radius. The first to use π with its present meaning was an
Welsh mathematician William Jones in 1706 when he states “3.14159 andc.
= π". Euler adopted the symbol in 1737 and it quickly became a
standard notation.

We conclude
with one further statistical curiosity about the calculation of π, namely
Buffon’s needle experiment. If we have a uniform grid of parallel lines, unit
distance apart and if we drop a needle of length k < 1 on the grid,
the probability that the needle falls across a line is 2k/π.
Various people have tried to calculate π by throwing needles. The most
remarkable result was that of Lazzerini (1901), who made 34080 tosses and got

π = 355/113

= 3.1415929

which,
incidentally, is the value found by Zu Chongzhi. This outcome is suspiciously
good, and the game is given away by the strange number 34080 of tosses. Kendall
and Moran comment that a good value can be obtained by stopping the experiment
at an optimal moment. If you set in advance how many throws there are to be
then this is a very inaccurate way of computing π. Kendall and Moran
comment that you would do better to cut out a large circle of wood and use a
tape measure to find its circumference and diameter.

Still on the
theme of phoney experiments, Gridgeman, in a paper which pours scorn on
Lazzerini and others, created some amusement by using a needle of carefully
chosen length k = 0.7857, throwing it twice, and hitting a line once.
His estimate for π was thus given by

2 cross0.7857
/ π = 1/2

from which
he got the highly creditable value of π = 3.1428. He was not being
serious!

It is almost
unbelievable that a definition of π was used, at least as an excuse, for a
racial attack on the eminent mathematician Edmund Landau in 1934. Landau had
defined π in this textbook published in Göttingen in that year by the, now
fairly usual, method of saying that π/2 is the value of x between 1
and 2 for which cos x vanishes. This unleashed an academic dispute which
was to end in Landau’s dismissal from his chair at Göttingen. Bieberbach, an
eminent number theorist who disgraced himself by his racist views, explains the
reasons for Landau’s dismissal:-

Thus the
valiant rejection by the Göttingen student body which a great mathematician,
Edmund Landau, has experienced is due in the final analysis to the fact that
the un-German style of this man in his research and teaching is unbearable to
German feelings. A people who have perceived how members of another race are
working to impose ideas foreign to its own must refuse teachers of an alien
culture.

G H Hardy
replied immediately to Bieberbach in a published note about the consequences of
this un-German definition of π

There are
many of us, many Englishmen and many Germans, who said things during the War
which we scarcely meant and are sorry to remember now. Anxiety for one’s own
position, dread of falling behind the rising torrent of folly, determination at
all cost not to be outdone, may be natural if not particularly heroic excuses.
Professor Bieberbach’s reputation excludes such explanations of his utterances,
and I find myself driven to the more uncharitable conclusion that he really
believes them true.

Not only in
Germany did π present problems. In the USA the value of π gave rise
to heated political debate. In the State of Indiana in 1897 the House of
Representatives unanimously passed a Bill introducing a new mathematical
truth.

Be it
enacted by the General Assembly of the State of Indiana: It has been found that
a circular area is to the square on a line equal to the quadrant of the
circumference, as the area of an equilateral rectangle is to the square of one
side.

The Senate
of Indiana showed a little more sense and postponed indefinitely the adoption
of the Act!

OPEN
QUESTIONS ABOUT THE NUMBER Π

1.      Does each of
the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in π?

2.      Brouwer’s
question: In the decimal expansion of π, is there a place where a thousand
consecutive digits are all zero?

3.      Is π
simply normal to base 10? That is does every digit appear equally often in its
decimal expansion in an asymptotic sense?

4.      Is π
normal to base 10? That is does every block of digits of a given length appear
equally often in its decimal expansion in an asymptotic sense?

5.      Is π normal

? That is does every block of digits of a given length appear equally often in
the expansion in every base in an asymptotic sense? The concept was introduced
by Borel in 1909.

6.      Another
normal question! We know that π is not rational so there is no point from
which the digits will repeat. However, if π is normal then the first
million digits 314159265358979… will occur from some point. Even if π is
not normal this might hold! Does it? If so from what point? Note: Up to 200
million the longest to appear is 31415926 and this appears twice.

As a
postscript, here is a mnemonic for the decimal expansion of π. Each
successive digit is the number of letters in the corresponding word.

How I want a
drink, alcoholic of course, after the heavy lectures involving quantum
mechanics. All of thy geometry, Herr Planck, is fairly hard…
:

3.14159265358979323846264…

A Chronology
of Pi (∏)

PRE COMPUTER CALCULATIONS OF Π

Mathematician

Date

Places

Comments

1

Rhind
papyrus

2000 BC

1

3.16045 (=
4(8/9)2)

2

Archimedes

250 BC

3

3.1418
(average of the bounds)

3

Vitruvius

20 BC

1

3.125 (= 25/8)

4

Chang Hong

130

1

3.1622 (=
√10)

5

Ptolemy

150

3

3.14166

6

Wang Fan

250

1

3.155555
(= 142/45)

7

Liu Hui

263

5

3.14159

8,

Zu
Chongzhi

480

7

3.141592920
(= 355/113)

9

Aryabhata

499

4

3.1416 (= 62832/2000)

10

Brahmagupta

640

1

3.1622 (=
√10)

11

Al-Khwarizmi

800

4

3.1416

12

Fibonacci

1220

3

3.141818

13

Madhava

1400

11

3.14159265359

14

Al-Kashi

1430

14

3.14159265358979

15

Otho

1573

6

3.1415929

16

Viète

1593

9

3.1415926536

17

Romanus

1593

15

3.141592653589793

18

Van Ceulen

1596

20

3.14159265358979323846

19

Van Ceulen

1596

35

3.1415926535897932384626433832795029

20

Newton

1665

16

3.1415926535897932

21

Sharp

1699

71

22

Seki Kowa

1700

10

23

Kamata

1730

25

24

Machin

1706

100

25

De Lagny

1719

127

Only 112
correct

26

Takebe

1723

41

27

Matsunaga

1739

50

28

von Vega

1794

140

Only 136
correct

29

Rutherford

1824

208

Only 152
correct

30

Strassnitzky,
Dase

1844

200

31

Clausen

1847

248

32

Lehmann

1853

261

33

Rutherford

1853

440

34

Shanks

1874

707

Only 527
correct

35

Ferguson

1946

620

General
Remarks:

A. In early
work it was not known that the ratio of the area of a circle to the square of
its radius and the ratio of the circumference to the diameter are the same.
Some early texts use different approximations for these two
"different" constants. For example, in the Indian text the Sulba
Sutras
the ratio for the area is given as 3.088 while the ratio for the
circumference is given as 3.2.

B.
Euclid gives in the Elements XII Proposition 2:

Circles are to one another as the squares on their diameters.

He makes no attempt to calculate the ratio.

COMPUTER CALCULATIONS OF Π

Mathematician

Date

Places

Type of
computer

Ferguson

Jan 1947

710

Desk
calculator

Ferguson,
Wrench

Sept 1947

808

Desk
calculator

Smith,
Wrench

1949

1120

Desk
calculator

Reitwiesner
et al.

1949

2037

ENIAC

Nicholson,
Jeenel

1954

3092

NORAC

Felton

1957

7480

PEGASUS

Genuys

Jan 1958

10000

IBM 704

Felton

May 1958

10021

PEGASUS

Guilloud

1959

16167

IBM 704

Shanks,
Wrench

1961

100265

IBM 7090

Guilloud,
Filliatre

1966

250000

IBM 7030

Guilloud,
Dichampt

1967

500000

CDC 6600

Guilloud,
Bouyer

1973

1001250

CDC 7600

Miyoshi,
Kanada

1981

2000036

FACOM
M-200

Guilloud

1982

2000050

Tamura

1982

2097144

MELCOM
900II

Tamura,
Kanada

1982

4194288

HITACHI
M-280H

Tamura,
Kanada

1982

8388576

HITACHI
M-280H

Kanada,
Yoshino, Tamura

1982

16777206

HITACHI
M-280H

Ushiro,
Kanada

Oct 1983

10013395

HITACHI
S-810/20

Gosper

Oct 1985

17526200

SYMBOLICS
3670

Bailey

Jan 1986

29360111

CRAY-2

Kanada,
Tamura

Sept 1986

33554414

HITACHI
S-810/20

Kanada,
Tamura

Oct 1986

67108839

HITACHI
S-810/20

Kanada,
Tamura, Kubo

Jan 1987

134217700

NEC SX-2

Kanada,
Tamura

Jan 1988

201326551

HITACHI
S-820/80

Chudnovskys

May 1989

480000000

Chudnovskys

June 1989

525229270

Kanada,
Tamura

July 1989

536870898

Chudnovskys

Aug 1989

1011196691

Kanada,
Tamura

Nov 1989

1073741799

Chudnovskys

Aug 1991

2260000000

Chudnovskys

May 1994

4044000000

Kanada,
Tamura

June 1995

3221225466

Kanada

Aug 1995

4294967286

Kanada

Oct 1995

6442450938

Kanada, Takahashi

Aug 1997

51539600000

HITACHI
SR2201

Kanada,
Takahashi

Sept 1999

206158430000

HITACHI
SR8000

General Remarks:

A.
Calculating π to many decimal places was used as a test for new computers
in the early days.

B.
There is an algorithm by Bailey, Borwein and Plouffe, published in 1996, which
allows the nth hexadecimal digit of π to be computed without the
preceeding n– 1 digits.

C.
Plouffe discovered a new algorithm to compute the nth digit of π in
any base in 1997.

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