An overview of Indian mathematics
It is without doubt that mathematics today owes a huge debt to the
outstanding contributions made by Indian mathematicians over many
hundreds of years. What is quite surprising is that there has been a
reluctance to recognise this and one has to conclude that many famous
historians of mathematics found what they expected to find, or perhaps
even what they hoped to find, rather than to realise what was so clear
in front of them.
We shall examine the contributions of Indian
mathematics in this article, but before looking at this contribution in
more detail we should say clearly that the "huge debt" is the beautiful
number system invented by the Indians on which much of mathematical
development has rested. Laplace put this with great clarity:-
The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.
We shall look briefly at the Indian development of the place-value
decimal system of numbers later in this article and in somewhat more
detail in the separate article Indian numerals. First, however, we go back to the first evidence of mathematics developing in India.
Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras
but research into the history of Indian mathematics has shown that the
essentials of this geometry were older being contained in the altar
constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita.
Also it has been shown that the study of mathematical astronomy in
India goes back to at least the third millennium BC and mathematics and
geometry must have existed to support this study in these ancient times.
The first mathematics which we shall describe in this article developed
in the Indus valley. The earliest known urban Indian culture was first
identified in 1921 at Harappa in the Punjab and then, one year later, at
Mohenjo-daro, near the Indus River in the Sindh. Both these sites are
now in Pakistan but this is still covered by our term "Indian
mathematics" which, in this article, refers to mathematics developed in
the Indian subcontinent. The Indus civilisation (or Harappan
civilisation as it is sometimes known) was based in these two cities and
also in over a hundred small towns and villages. It was a civilisation
which began around 2500 BC and survived until 1700 BC or later. The
people were literate and used a written script containing around 500
characters which some have claimed to have deciphered but, being far
from clear that this is the case, much research remains to be done
before a full appreciation of the mathematical achievements of this
ancient civilisation can be fully assessed.
We often think of Egyptians and Babylonians as being the height of
civilisation and of mathematical skills around the period of the Indus
civilisation, yet V G Childe in New Light on the Most Ancient East (1952) wrote:-
India confronts Egypt and Babylonia by the 3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture.
We do know that the Harappans had adopted a uniform system of weights
and measures. An analysis of the weights discovered suggests that they
belong to two series both being decimal in nature with each decimal
number multiplied and divided by two, giving for the main series ratios
of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several
scales for the measurement of length were also discovered during
excavations. One was a decimal scale based on a unit of measurement of
1.32 inches (3.35 centimetres) which has been called the "Indus inch".
Of course ten units is then 13.2 inches which is quite believable as the
measure of a "foot". A similar measure based on the length of a foot is
present in other parts of Asia and beyond. Another scale was discovered
when a bronze rod was found which was marked in lengths of 0.367
inches. It is certainly surprising the accuracy with which these scales
are marked. Now 100 units of this measure is 36.7 inches which is the
measure of a stride. Measurements of the ruins of the buildings which
have been excavated show that these units of length were accurately used
by the Harappans in construction.
It is unclear exactly what caused the decline in the Harappan
civilisation. Historians have suggested four possible causes: a change
in climatic patterns and a consequent agricultural crisis; a climatic
disaster such flooding or severe drought; disease spread by epidemic; or
the invasion of Indo-Aryans peoples from the north. The favourite
theory used to be the last of the four, but recent opinions favour one
of the first three. What is certainly true is that eventually the
Indo-Aryans peoples from the north did spread over the region. This
brings us to the earliest literary record of Indian culture, the Vedas
which were composed in Vedic Sanskrit, between 1500 BC and 800 BC. At
first these texts, consisting of hymns, spells, and ritual observations,
were transmitted orally. Later the texts became written works for use
of those practicing the Vedic religion.
The next mathematics of importance on the Indian subcontinent was associated with these religious texts. It consisted of the Sulbasutras
which were appendices to the Vedas giving rules for constructing
altars. They contained quite an amount of geometrical knowledge, but the
mathematics was being developed, not for its own sake, but purely for
practical religious purposes. The mathematics contained in the these
texts is studied in some detail in the separate article on the Sulbasutras.
The main Sulbasutras were composed by Baudhayana (about 800 BC), Manava (about 750 BC), Apastamba (about 600 BC), and Katyayana
(about 200 BC). These men were both priests and scholars but they were
not mathematicians in the modern sense. Although we have no information
on these men other than the texts they wrote, we have included them in
our biographies of mathematicians. There is another scholar, who again
was not a mathematician in the usual sense, who lived around this
period. That was Panini
who achieved remarkable results in his studies of Sanskrit grammar.
Now one might reasonably ask what Sanskrit grammar has to do with
mathematics. It certainly has something to do with modern theoretical
computer science, for a mathematician or computer scientist working with
formal language theory will recognise just how modern some of Panini's ideas are.
Before the end of the period of the Sulbasutras, around the middle of the third century BC, the Brahmi numerals had begun to appear.
Here is one style of the Brahmi numerals..
These are the earliest numerals which, after a multitude of changes, eventually developed into the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 used today. The development of numerals and place-valued number systems are studied in the article Indian numerals.
The Vedic religion with its sacrificial rites began to
wane and other religions began to replace it. One of these was Jainism,
a religion and philosophy which was founded in India around the 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.
You can see a separate article about Jaina mathematics.
If the Vedic religion gave rise to a study of mathematics for constructing sacrificial altars, then it was Jaina cosmology which led to ideas of the infinite in Jaina mathematics. Later mathematical advances were often driven by the study of astronomy. Well perhaps it would be more accurate to say that astrology formed the driving force since it was that "science" which required accurate information about the planets and other heavenly bodies and so encouraged the development of mathematics. Religion too played a major role in astronomical investigations in India for accurate calendars had to be prepared to allow religious observances to occur at the correct times. Mathematics then was still an applied science in India for many centuries with mathematicians developing methods to solve practical problems.
You can see a separate article about Jaina mathematics.
If the Vedic religion gave rise to a study of mathematics for constructing sacrificial altars, then it was Jaina cosmology which led to ideas of the infinite in Jaina mathematics. Later mathematical advances were often driven by the study of astronomy. Well perhaps it would be more accurate to say that astrology formed the driving force since it was that "science" which required accurate information about the planets and other heavenly bodies and so encouraged the development of mathematics. Religion too played a major role in astronomical investigations in India for accurate calendars had to be prepared to allow religious observances to occur at the correct times. Mathematics then was still an applied science in India for many centuries with mathematicians developing methods to solve practical problems.
Yavanesvara,
in the second century AD, played an important role in popularising
astrology when he translated a Greek astrology text dating from 120 BC.
If he had made a literal translation it is doubtful whether it would
have been of interest to more than a few academically minded people. He
popularised the text, however, by resetting the whole work into Indian
culture using Hindu images with the Indian caste system integrated into
his text.
By about 500 AD the classical era of Indian mathematics began with the work of Aryabhata.
His work was both a summary of Jaina mathematics and the beginning of
new era for astronomy and mathematics. His ideas of astronomy were truly
remarkable. He replaced the two demons Rahu, the Dhruva Rahu which
causes the phases of the Moon and the Parva Rahu which causes an eclipse
by covering the Moon or Sun or their light, with a modern theory of
eclipses. He introduced trigonometry in order to make his astronomical
calculations, based on the Greek epicycle theory, and he solved with
integer solutions indeterminate equations which arose in astronomical
theories.
Aryabhata
headed a research centre for mathematics and astronomy at Kusumapura in
the northeast of the Indian subcontinent. There a school studying his
ideas grew up there but more than that, Aryabhata
set the agenda for mathematical and astronomical research in India for
many centuries to come. Another mathematical and astronomical centre was
at Ujjain, also in the north of the Indian subcontinent, which grew up
around the same time as Kusumapura. The most important of the
mathematicians at this second centre was Varahamihira who also made important contributions to astronomy and trigonometry.
The main ideas of Jaina mathematics, particularly
those relating to its cosmology with its passion for large finite
numbers and infinite numbers, continued to flourish with scholars such
as Yativrsabha. He was a contemporary of Varahamihira and of the slightly older Aryabhata.
We should also note that the two schools at Kusumapura and Ujjain were
involved in the continuing developments of the numerals and of
place-valued number systems. The next figure of major importance at the
Ujjain school was Brahmagupta
near the beginning of the seventh century AD and he would make one of
the most major contributions to the development of the numbers systems
with his remarkable contributions on negative numbers and zero. It is a
sobering thought that eight hundred years later European mathematics
would be struggling to cope without the use of negative numbers and of
zero.
These were certainly not Brahmagupta's
only contributions to mathematics. Far from it for he made other major
contributions in to the understanding of integer solutions to
indeterminate equations and to interpolation formulas invented to aid
the computation of sine tables.
The way that the contributions of these mathematicians were prompted by a
study of methods in spherical astronomy is described in [25]:-
The Hindu astronomers did not possess a general method for solving problems in spherical astronomy, unlike the Greeks who systematically followed the method of Ptolemy, based on the well-known theorem of Menelaus. But, by means of suitable constructions within the armillary sphere, they were able to reduce many of their problems to comparison of similar right-angled plane triangles. In addition to this device, they sometimes also used the theory of quadratic equations, or applied the method of successive approximations. ... Of the methods taught by Aryabhata and demonstrated by his scholiast Bhaskara I, some are based on comparison of similar right-angled plane triangles, and others are derived from inference. Brahmagupta is probably the earliest astronomer to have employed the theory of quadratic equations and the method of successive approximations to solving problems in spherical astronomy.
Before continuing to describe the developments through the classical
period we should explain the mechanisms which allowed mathematics to
flourish in India during these centuries. The educational system in
India at this time did not allow talented people with ability to receive
training in mathematics or astronomy. Rather the whole educational
system was family based. There were a number of families who carried the
traditions of astrology, astronomy and mathematics forward by educating
each new generation of the family in the skills which had been
developed. We should also note that astronomy and mathematics developed
on their own, separate for the development of other areas of knowledge.
Now a "mathematical family" would have a library which
contained the writing of the previous generations. These writings would
most likely be commentaries on earlier works such as the Aryabhatiya of Aryabhata.
Many of the commentaries would be commentaries on commentaries on
commentaries etc. Mathematicians often wrote commentaries on their own
work. They would not be aiming to provide texts to be used in educating
people outside the family, nor would they be looking for innovative
ideas in astronomy. Again religion was the key, for astronomy was
considered to be of divine origin and each family would remain faithful
to the revelations of the subject as presented by their gods. To seek
fundamental changes would be unthinkable for in asking others to accept
such changes would be essentially asking them to change religious
belief. Nor do these men appear to have made astronomical observations
in any systematic way. Some of the texts do claim that the computed data
presented in them is in better agreement with observation than that of
their predecessors but, despite this, there does not seem to have been a
major observational programme set up. Paramesvara
in the late fourteenth century appears to be one of the first Indian
mathematicians to make systematic observations over many years.
Mathematics however was in a different position. It was only a tool used
for making astronomical calculations. If one could produce innovative
mathematical ideas then one could exhibit the truths of astronomy more
easily. The mathematics therefore had to lead to the same answers as had
been reached before but it was certainly good if it could achieve these
more easily or with greater clarity. This meant that despite
mathematics only being used as a computational tool for astronomy, the
brilliant Indian scholars were encouraged by their culture to put their
genius into advances in this topic.
A contemporary of Brahmagupta who headed the research centre at Ujjain was Bhaskara I who led the Asmaka school. This school would have the study of the works of Aryabhata as their main concern and certainly Bhaskara was commentator on the mathematics of Aryabhata. More than 100 years after Bhaskara lived the astronomer Lalla, another commentator on Aryabhata.
The ninth century saw mathematical progress with scholars such as Govindasvami, Mahavira, Prthudakasvami, Sankara, and Sridhara. Some of these such as Govindasvami and Sankara were commentators on the text of Bhaskara I while Mahavira was famed for his updating of Brahmagupta's
book. This period saw developments in sine tables, solving equations,
algebraic notation, quadratics, indeterminate equations, and
improvements to the number systems. The agenda was still basically that
set by Aryabhata and the topics being developed those in his work.
The main mathematicians of the tenth century in India were Aryabhata II and Vijayanandi,
both adding to the understanding of sine tables and trigonometry to
support their astronomical calculations. In the eleventh century Sripati and Brahmadeva were major figures but perhaps the most outstanding of all was Bhaskara II
in the twelfth century. He worked on algebra, number systems, and
astronomy. He wrote beautiful texts illustrated with mathematical
problems, some of which we present in his biography, and he provided the
best summary of the mathematics and astronomy of the classical period.
Bhaskara II may be considered the high point of Indian mathematics but at one time this was all that was known [26]:-
For a long time Western scholars thought that Indians had not done any original work till the time of Bhaskara II. This is far from the truth. Nor has the growth of Indian mathematics stopped with Bhaskara II. Quite a few results of Indian mathematicians have been rediscovered by Europeans. For instance, the development of number theory, the theory of indeterminates infinite series expressions for sine, cosine and tangent, computational mathematics, etc.
Following Bhaskara II
there was over 200 years before any other major contributions to
mathematics were made on the Indian subcontinent. In fact for a long
time it was thought that Bhaskara II
represented the end of mathematical developments in the Indian
subcontinent until modern times. However in the second half of the
fourteenth century Mahendra Suri wrote the first Indian treatise on the astrolabe and Narayana wrote an important commentary on Bhaskara II,
making important contributions to algebra and magic squares. The most
remarkable contribution from this period, however, was by Madhava who invented Taylor series and rigorous mathematical analysis in some inspired contributions. Madhava was from Kerala and his work there inspired a school of followers such as Nilakantha and Jyesthadeva.
Some of the remarkable discoveries of the Kerala mathematicians are described in [26]. These include: a formula for the ecliptic; the Newton-Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier's
formula for the circumradius of a cyclic quadrilateral. Of particular
interest is the approximation to the value of π which was the first to
be made using a series. Madhava's result which gave a series for π, translated into the language of modern mathematics, reads
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, x - y = b, xy = c, x2 + y2 = d, x2 - y2 = e, x3 + y3 = f, and x3 - y3 = g.
For each case, Citrabhanu gave an explanation and justification of his
rule as well as an example. Some of his explanations are algebraic,
while others are geometric. See [12] for more details.
Now we have presented the latter part of the history
of Indian mathematics in an unlikely way. That there would be
essentially no progress between the contributions of Bhaskara II and the innovations of Madhava,
who was far more innovative than any other Indian mathematician
producing a totally new perspective on mathematics, seems unlikely. Much
more likely is that we are unaware of the contributions made over this
200 year period which must have provided the foundations on which Madhava built his theories.
Our understanding of the contributions of Indian mathematicians has
changed markedly over the last few decades. Much more work needs to be
done to further our understanding of the contributions of mathematicians
whose work has sadly been lost, or perhaps even worse, been ignored.
Indeed work is now being undertaken and we should soon have a better
understanding of this important part of the history of mathematics.
