A Brief History of Ancient Indian
Mathematics
(India’s Romance with Numbers)
Rajen Barua
It is without doubt that mathematics, the science of numbers, today owes a huge debt to the
outstanding contributions made by Indian mathematicians over many hundreds of
years. From zero to geometry, Indian mathematicians made some great historical achievements. Frankly
speaking, without the Indian numerals, mathematics, as we know it today, would
simply not exist. This article is to focus on some of these achievements and to
show that mathematics was not only very much rooted in Indian soil but also
that ancient Indians had great romance with mathematics.
Indian mathematics may be said to have
started with the Vedic rituals which required knowledge of geometry for accurate
construction of Vedic altars. It developed further under the Jain and the
Buddhist scholars who pioneered some phenomenal ground level achievements. It
is now generally admitted that the Indian system of numbers has its roots firmly planted in India
and that it is the Indians who first invented and used the decimal place value
system including the use of the zero. The novel Indian numerals were
subsequently adopted by the Arabs, and eventually became known to Europe as
Arabic numerals. The ancient Indians provided a unique, useful, flexible and
intuitive model for the world to use.
The
great French Mathematician, Pascal was one who appreciated the contributions of
the Indians and he put this with great clarity, when he commented, “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.”
India has a long tradition, both historical and
mythical, of its fascination with numbers. In ancient India, mathematics was considered as one of the
highest sciences. There is a statement in the Vedanga Jyotisa, which proclaims, "As are the crests of a peacock, as are the gem-stones of a snake,
placed on the highest place of the body, the forehead, so is mathematics
(Ganita) the head of all Vedah and shastras." The quotation suggests a
reverent, almost elitist concept of mathematics in ancient India. In another
mythical statement we find, “What is the
use of much speaking. Whatever object exists in this moving and nonmoving
world, can not be understood without the base of Ganita (Mathematics)”.
Three thousands years later, Gaileo would realise the same when he
said, “It (the universe) is written in
the language of mathematics, and its characters are triangles, circles, and
other geometric figures, without which it is humanly impossible to understand a
single word; without these, one is wandering about in a dark labyrinth.”
The tradition of mathematics in India, in fact,
started much earlier of which we have solid historical facts in the Indus
valley which was associated with the Harappan civilization established around
2,500 B.C. We do know
that the Harappans had adopted a uniform system of weights and measures. An
analysis of the weights discovered suggests that they are decimal in nature,
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 centimeters) 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". Another scale was discovered
when a bronze rod was found which was marked in lengths of 0.367 inches. It is
surprising to see 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 not known fully if the Harappan knowledge in mathematics was continued in
later Vedic period. However we do know that from ancient time, Indian minds
fascinated with higher numbers. While the Greeks had no terminology for
denominations above a myriad (104) and the Romans above millie (103),
ancient Indians dealt freely with
no less than eighteen denominations. We know from a record of an early Buddhist
work in 5th century BC, how the prince Gautama Buddha correctly
recited the counting beyond the koti
on the centesimal scale: “Hundred kotis
are called ayuta, hundred ayutas niyuata, hundred niyutas kankara, hundred
kankaras vivara, hundred vivaras ksobhya, hundred ksobhyas vivaha, hundred
vivahas ustanga, hundred ustangas babula, hundred babulas nagabala, hundred
nagabalas tithi lambha and so on upto another twelve terms ending with
tallaksana. (Thus one Tallaksna is 1053).
Examples of geometric knowledge (rekha-ganit) are to be
found in the Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba
(400 BC). The term Sulvasutra
means “the rules of the chord”; it is the name given to the supplements of the Kalpasutras
which explain the construction of sacrificial Vedic altars. A statement
in Baudhayana's Sulvasutra runs
,”In a Deerghchatursh
(Rectangle) the Chetra (Square) of Rajju (hypotenuse) is
equal to sum of squares of Parshvamani (base) and Triyangmani (perpendicular).” This is what is commonly
known as the Pythagoras theorem.
This shows that Vedic Indians had knowledge of the Pythagoras theorem as
early as 8th century B.C. Apasthamba's sutra provides a value
for the square root of 2 that is accurate to the fifth decimal place. Apasthamba
also looked at the problems of squaring a circle, dividing a segment into seven
equal parts, and a solution to the general linear equation.
However we donot find any proof of any of the theorems in Vedic
mathematics, just the statements. This may be because, the involved mathematics
was considered a sacred and secret knowledge just for the Vedic rituals
reserved for the priests only. This has been one of the main problems for
further development on Indian science during the Vedic Brahmanic age. As a
result, none of these geometrical constructions appeared in
any subsequent Indian literature, and later mathematicians did not carry these
discussions to any higher level like what the Greeks did. As
Gordon Childe, the famous historian rightly puts it: while the Greeks were free to speculate on ‘facts of common experience’
and ‘the practice of the craft’, the Vedic Indians were restricted by their
‘inheriting from the Bronze age the sacred hymns of Veda and ritual manuals
verbally remembered’. In fact the Indians could never got rid of the Vedas
completely which prevented them, throughout the ages, from exploring ‘secular’
scientific speculation to higher limits.
Thus it was mainly under the religious and
philosophical impulses of the Jains and the Buddhists, that Indian mind learned
to speculate outside the Vedas, and science and mathematics got some freedom. In ancient India, mathematics was a
very lively passion. In Sanskrit, Ganita
literally means the science of calculations which were generally done on a
board (pati) with a piece of chalk or
on sand (dhuli) spread on the ground.
Thus the terms Pati-Ganita (science
of calculations on the borad) and Dhuli-Karma
(dust work) came to be used for higher mathematics. In
ancient Buddhist literature we find mention of three classes of Ganita (1) Mudra – finger arithmetic (2)
Ganana – mental arithmetic (3) Samkhyana – higher mathematics in
general.
These Jains and Buddhist scholars worked on
problems such as number theory, cubic equations, quadratic equations, and
statistics. They also had an understanding of advanced ideas such as that of
infinity. The Jains were the first to discard the idea that all infinites were the
same or equal. They recognized four different types of infinities: infinite in
length (one dimension), infinite in area (two dimensions), infinite in volume (three
dimensions), and infinite perpetually (infinite number of dimensions). Jain texts from the 6th
C BC such as the Surya Pragyapti describe ellipses. Buddhist literature
also demonstrates an awareness of indeterminate and infinite numbers. Numbers
were deemed to be of three types: Sankheya (countable), Asankheya
(uncountable) and Anant (infinite). The Buddhist philosophical formulations
concerning Shunya - i.e. emptiness or the void may have facilitated in
the introduction of the concept of zero. While the zero (bindu) as an
empty place holder in the place-value numeral system appears much earlier,
algebraic definitions of the zero and it's relationship to mathematical
functions appear in the mathematical treatises of Brahmagupta in the 7th C AD.
The early Jainas seem to
have great liking for the subject of combinations and permutations. Mahabira,
the founder of Jainism, was himself a great mathematician. In the Bhagawati
sutra are set forth simple problems such as finding the number of
combinations that can be obtained from a given number of fundamental
philosophical categories taken one at a time, two at a time, three at a time or
more at a time. The Jaina commentator Silanka has quoted three rules regarding
permutations and combinations. The Jains were the first to conceive of
transfinite numbers, a concept, which was brought to Europe by Cantor in the
late 19th century. The two thousand year old Jaina literature may hold valuable
clues to the very nature of mathematics. This is one area where further
research could prove very fruitful.
The works of the early Jain and Buddhist
scholars were later summarized and expanded by Aryabhatta (476-550), the most important ancient mathematician of
India. Aryabhatta headed the
classical era of Indian mathematics. He helped to ignite a new era in
mathematics, which in turn spurred on other sciences, such as astronomy. Among
his many accomplishments were the introduction of the concept of trigonometry,
the most precise estimation of π (the ratio of the circumference to the
diameter of a circle) up to that date (3.1416), and an accurate estimation of a
solar year. His
calculations on the circumference of the earth (62832 miles) and the length of
the solar year (within about 13 minutes of the modern calculation) were
remarkably close approximations. In making such calculations, Aryabhatta had
to solve several mathematical problems that had not been addressed before
including problems in algebra (bij-ganit) and trigonometry. In the
course of developing a precise mapping of the lunar eclipse, Aryabhatta was
obliged to introduce the concept of infinitesimals - i.e. tatkalika gati to
designate the infinitesimal, or near instantaneous motion of the moon, and
express it in the form of a basic differential equation. It is worth mentioning that Roots of
the Modern Trignometry lie in the book titled Surya Siddhanta . It
mentions Zia (Sine), Kotizia (Cosine) etc. Please note that
the word (Zia) changed to "Jaib" in Arab first. The translation of
Jaib in Latin was done as "Sinus". And this "Sinus" became
"Sine" later on. The word Trigonometry also is derived from the
Indian word Trikonomiti, Trikono meaning a Triangle, (modern day Tribhuja).
Bhaskar I continued where Aryabhatta
left off, and discussed in further detail topics such as the longitudes of the
planets; conjunctions of the planets with each other and with bright stars;
risings and settings of the planets; and the lunar crescent. Again, these
studies required still more advanced mathematics and Bhaskar I expanded
on the trigonometric equations provided by Aryabhatta, and like Aryabhatta
correctly assessed π (pi) to be an irrational number. Amongst his most
important contributions was his formula for calculating the sine function which
was 99% accurate. He also did pioneering work on indeterminate equations and
considered for the first time quadrilaterals with all the four sides unequal
and none of the opposite sides parallel.
Another
important astronomer/mathematician was Varahamira (6th C, Ujjain) who
compiled previously written texts on astronomy and made important additions to Aryabhatta's
trigonometric formulas. His works on permutations and combinations
complemented what had been previously achieved by Jain mathematicians and
provided a method of calculation of n-C-r that closely resembles
the much more recent Pascal's Triangle. In the 7th century, Brahmagupta
(born 598) did important work in enumerating the basic principles of
algebra. In 628 he wrote his Brahma-sphuta-siddhanta
(“The Revised System of Brahma”). In addition to listing the algebraic
properties of zero, he also listed the algebraic properties of negative numbers
and used these signs for addition,
subtraction and multiplication (+, -, x). He was the first to postulate that, “The
multiplication of a positive number with a negative number comes out to be a
negative number. Further when a positive number is divided by a negative number
or a negative number is divided by a positive number the result is a negative
number.”
His work on solutions to quadratic
indeterminate equations anticipated the work of Euler and Lagrange. It is a
sobering thought that even eight hundred years later European mathematics would
be struggling to cope with the use of negative numbers and of zero. More than that, he made other major contributions to the
understanding of integer solutions to indeterminate equations and to
interpolation formulas invented to aid the computation of sine tables. Although almost all ancient countries used
quantities of unknown values for solution of problems, the expansion of Algebra
(Biz Ganit) became possible only when Indians realized that all the
calculations of Numerical Mathematics could be done by notations, and for the
first time used Sanskrit Alphabet to denote unknown quantities.
Brahmagupta is also credited with
the following:
- To give the general solution to the quadratic equation ax2 + bx + c = 0 in the form : x = [-b ± (b2 - 4ac)1/2]/2a
- To give an alternate proof of the Pythagorean Theorem by calculating the same area in two different ways and then canceling out terms to get a² + b² = c².
Developments
also took place in applied mathematics such as in creation of trigonometric tables and measurement units. Yativrsabha's
work Tiloyapannatti (6th C) gives various units for measuring
distances and time and also describes the system of infinite time measures
Aryabhatta's equations were elaborated on by Manjula
(10th C) and Bhaskaracharya (12th C) who derived the differential of the
sine function. Later mathematicians used their intuitive understanding of
integration in deriving the areas of curved surfaces and the volumes enclosed
by them.
Between the 7th C and the 11th C, Indian
numerals developed into their modern form, and along with the symbols denoting
various mathematical functions eventually became the foundation stones of
modern mathematical notation.
The study of original mathematics
in India slowed down after 8th century. This was also the time when Buddhism started to decline in India.
The onslaught of the Islamic invasions in the twelfth century gave a great blow
to Buddhism and basically expelled Buddhism out of India. Since that time,
India may be said to undergo a period of dark age. During this period, secular
studies in mathematics suffered greatly. During this time, the main center of
studies in mathematics also gradually shifted to south India
But this was also the time when
Indian mathematical texts were increasingly being translated into Arabic and
Persian. The last notable Indian mathematician may be said to be Bhaskaracharya who came from a long-line of mathematicians
and was head of the astronomical observatory at Ujjain. He left several
important mathematical texts including the Lilavati, Bijaganita and the
Siddhanta Shiromani, an astronomical text. He is however more popularly
known today for his book, Lilavati
which is a unique book which shows how mathematics was brought to the reach of
the common people. It
is a collection of worked out examples of arithmetic, algebra, geometry, and
mensuration. The language is written in Sanskrit verse, and the level of
mathematics ranges between high school algebra to pre-Calculus. In its time, it
represented the height of 12th-century mathematics. The problems are
generally addressed to one Lilavati,
traditionally taken to be either his wife or his daughter. Reading Lilavati, any reader will find that
learning mathematics can be fun which also flourishes in wonder.
A
typical problem reads, “O deer eyed one!
Tell me if one sixth of the number of bees in a colony entered a jasmine flower
tree, one-third went to kadamba tree, one fourth flew to a mango tree,
one-fifth went to a tree blooming with sampaka flowers, one-thirtieth went to a
beautiful bed of lotuses bloomed by the Sun’s rays and if the remaining one bee
was roving about, how many total bees were there in the colony?”
This
is a problem of unknown quantity which is solved by now a days by assuming X
for the unknown quantity. In this case, the problem resolves to
(X)
– (X/6) – (X/3) – (X/4) – (X/5) – (X/30) = 1; Or, (X/60)
=1; Or, X = 60
There were also problems
that excite real passion for mathematics. Note the following verse: Whilst making love, a necklace broke.
A row of pearls mislaid.
One third fell to the floor.
One fifth upon the bed.
The young woman saved one sixth of them.
One tenth were caught by her lover.
If six pearls remained upon the string
How many pearls were there altogether?
This is another problem of finding the unknown number. The modern way of solving is to assume X for the unknown number. Then the problem can be written as:
(X) - (X/3) – (X/5) – (X/6) – (X/10) = 6; or X (30-10-6-5-3) / 30 = 6; or (6X) / 30 = 6; or X = 30
Another
problem describes as “A flock of swans
contained total x2 members. As clouds gathered, (10x) of these went
to Manasa lake, and (1/8 x2) flew away to a garden. The remaining
three couples played about in the water. O young woman, how many swans were
there in that lake full of beautiful lotuses?”
This is a problem of
quadratic equation which can be written as:
(X2) – (10X) - (X2/8) = 6 ; the problem is solved
for X = 31; which makes the number of swans to be X2=961
Another
problem that deals higher mathematics runs like this: ”Dear Lilavaty, Suppose different kinds of Chatnies are made by
mixing 1, 2, 3, 4, 5 or 6 at a time from six substances which are respectively
sweet, bitter, astringent, sour, salty and hot. O, my pretty mathematician,
tell me how many different Chatnies can be prepared from these all?”
This
a problem of Combination which can be solved by the formula for Combination of
n things taken 1, 2, 3 … r things at a time. The formula is :
n-C-r = n! / [r!(n-r)!] (Where
n is the total number of things taken r things at a time)
The
solution is: = n!/0!(n!), n!/1!(n-1)!, n!/2!(n-2)!, n!/3!(n-3)!,….n!/n!(n-n)!
=
6!/0!(6!), 6!/1!(5!), 6!/2!(4!), 6!/3!(3!), 6!/4!(2!), 6!/5!(1!), 6!/6!(0!)
=
1, 6, 15, 20, 15, 6, 1 = 64, the total number of different chatnies that can be made.
[In
the above, [n! is called factorial n ; example, Factorial 3!=3.2.1=6] The answers also can be found from
modern day Pascels Triangle which was used by ancient Indian mathematicians and
was known as Khandmeru.
The lucid, scholarly and literary presentation in Lilavaty has attracted several cultural
areas. The graphic descriptions of a drove
of swans; a flock of elephants; a colony of bees; the attack of a snake by a
domesticated peacock; sinking of a lotus in water owing to strong wind; and
many others were to train the students not only in mathematics but also in
appreciation of nature. No wonder that Lilavati has not only
been used widely in India in the medieval times as a standard base text-book
for about 800 years, but it was commented and translated into several languages
of the world.
Besides that, like poetry, mathematics has a
beauty and truth, and can be enjoyed as such. Plato once said, “Arithmetic
has a very great elevating effect, compelling the soul to reason about abstract
numbers refusing to be satisfied visible and tangible objects.” The joy of
mathematics is similar to the experience of discovering something new for the
first time. In Lilavaty, Bhaskara showed that he was not only a great
mathematician but was also a great poet who may be compared to
Omar Khayyam. His conclusion to Lilavati states: “Joy and happiness is indeed ever increasing in this world for those
who have Lilavati clasped to their throats.” Overall, Bhaskaracharya’s Lilavati proves the point that Indians
had great romance with mathematics and that it is very much ingrained in the
Indian culture. The creation of a book like Lilavaty
was possible in India because of its long tradition of culture with mathematics;
one will not find such book in any other civilization.
References
are made to the following book while preparing this article.
1.
History
of Hindu Mathematics-B.B. Datta & A.N. Singh
2.
Geometry
According to Sulba Sutra- Dr. R.P. Kulkarni
3.
Geometry
in Ancient and Medieval India – Dr. T.A.S. Amma
4.
Vedic
Mathematics in School- J.T. Glover.
5.
Aryabhatiya
of Aryabhatta-Edited by.- Walter Eugene Clark
6.
Surya-Siddhanta
– Edited by- Phanindra Lal Ganguli
7.
Lilavati-
Edited by- Krishnaji Shankar Patwardhan
8.
Buddhism
and Science – Buddhadasa P. Kirthisinghe
9.
India’s
Contribution to the West – Dr. P. Priyadarshi
10.
What
Happened in History – Gordon Childe
11.
The
Story of Mathematics – Richard Mankiewicz
12.
Joy of
Mathematics – Theoni Pappas
13.
Mathematics
for the Non Mathematicians – Morris Kline
14.
Mathematics
for Millions – Lancelot Hogben
15.
God
Created the Integers – Stephen Hawkins
16.
Universal
History of Numbers – Georges Ifrah
17.
History
of Mathematics – D.E. Smith
18.
History
of Mathematics – Howard Eves
19.
History
of Mathematics – David M. Burton
20.
The
World of Mathematics – James R Newman
21.
Men of
Mathematics – E.T. Bell
22. Makers of Mathematics – Stuart
Hollingdale
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