**History of Mathematics -
The Indian Contribution**

The Contribution to Mathematics by

1.
Zero and the place-value
notation for numbers

2.
Vedic Mathematics and
arithmetical operations

3.
Geometry of the Sulba
Sutras

4.
Jaina contribution to
Fundamentals of numbers

5.
The anonymous Bakshali manuscript

6.
Astronomy

7.
Classical contribution to
Indeterminate Equations and Algebra

8.
Indian Trigonometry

9.
Kerala contribution to
Infinite Series and Calculus.

10.
Modern Contribution:
Srinivasa Ramanujan onwards

**Section
1: Zero and the Place-Value Notation**

The number zero is the subtle gift of the Hindus of antiquity
to mankind. The concept itself was one of the most significant inventions in
the ascent of Man for the growth of
culture and civilization. To it must be credited the enormous usefulness of its counterpart,
the place value system of expressing all numbers with just ten symbols. And to
these two concepts we owe all the arithmetic and mathematics based upon them,
the great ease which it has lent to all computations for two millenia and the
binary system which now lies at the
foundation of communicating with computers. Already in the first three
centuries A.D.. the Hindu ancients were using a decimal positional system, that
is, a system in which numerals in different positions represent different
numbers and in which one of the ten symbols used was a fully functional zero. They called it '*Sunya'*. The word and its meaning ‘void’ were obviously
borrowed from its use in philosophical literature. Though the Babylonians used
a special symbol for zero as early as the 3^{rd} century B.C. , they
used it only as a place holder; they did not have the concept of zero as an
actual value. It appears the Maya civilisation of

1.
A notation for powers of 10 upto the power 17 was already in
use even from vedic times. Single words have been used to denote the powers of
the number 10. The numbers one, ten, hundred, thousand, ten thousand, … are
given by the sequence of words in the list: *eka, dasa, Sata, sahasra, ayuta, laksha,
prayuta, koTi, arbuda, abja, kharva, nikharva, mahA-padma, Sankha, jaladhi,
antya, mahASankha, parArdha. * Thus
the decimal system was in the culture even in
the early part of the first millenium B.C. . The *Yajurveda*,
in its description of rituals and the *mantras*
employed therein, the *Mahabharata* and
the Ramayanaa in their descriptions of statistics and measurements, used all
these words, with total abandon.

2.
Counting boards with columns representing units and tens were
in use from very ancient times. The numberless content of an empty column in course of time was symbolized to be ‘nothing’.

3.
The thriving activity
in astrology, astronomy, navigation and business during the first few centuries A.D. naturally looked forward to a superior
numerical system that lent itself to complicated calculations.

4.
Distinct symbols for the numbers 1 to 9 already existed and
the counting system used the base 10 in
all its secular, religious and ritual activities. Compare this with the
Babylonian numeration which had only three figures, one for 1, one for 10, and
one for 100, so that a number, say, 999, would require 27
symbols, namely, nine of each of the symbols.

Of these, the first and fourth factors are probably unique to Hindu culture and
contributed most to the thought process that led to the decimal place value notation as well as zero having a value. When exactly the invention of this most modest of all numerals
took place, we do not know. The first time it reached Europe was during the
Moorish invasion of

How the *Sunya*
of the Hindus became the Zero of the modern world is interesting. The '*Sunya*'
of Sanskrit became the Arabic ‘*sifr*’ which means empty space. In
medieval Latin it manifested as ‘*ciphra’* , then in
middle English as ‘*siphre*’, in English as ‘*cypher*’ and in American as ‘*cipher*’. In the middle ages, the word ‘*ciphra*’ evolved to stand for the whole
system. In the wake of this general
meaning, the Latin ‘*zephirum*’ came to be used to denote the *Sunya*.
And that entered English finally as ‘*zero*’.
In medieval *Sunya* to be a creation of the devil! As a result ‘*ciphra*’ came to mean a secret code. From this came
‘deciphering’, the resolution of a code!

**Section 2. Vedic Mathematics and
arithmetical operations**

Vedic Mathematics provides
an original and refreshing approach to subjects which are usually dismissed as
mechanical and tedious. Bharati Krishna Tirtha who published his reconstruction of Vedic Mathematics in
1965, maintains that there are 16 aphorisms and 13 secondary aphorisms which
forms his base of the so-called Vedic Mathematics. Though the origins of Vedic Mathematics have
not yet been historically established, if nothing else, it provides tremendous
insights into the place-value system of numbers
without which it would not work. It is amazing that Vedic Mathematics
does not require of cramming of multiplication tables beyond 5 x 5. One can
improvise all the necessary multiplication tables for oneself and with the aid of the relevant Vedic
formulae get the required products very
easily, speedily, and correctly, almost immediately. The formulae can be used to evaluate
determinants, solve simultaneous linear equations, evaluate logarithms and
exponentials. Vedic Mathematics recognises that
any algebraic polynomial may be expressed in terms of a positional
notation without specifying the base. The same algorithmic scheme as applied to
arithmetical operations will easily apply to algebraic problems. And this brings it to the Modern Algebra of
Polynomials. It is difficult, in a historical introduction like this to get
into the details of Vedic Mathematics.
Suffice it to say that with today's over-dependence on calculators for
even simple arithmetical computations, the Vedic methods have great pedagogical
value and, through their revival, the skills of mental arithmetic may not be
lost for posterity.

**Section 3. Geometry of the Sulba Sutras **

Hailing from the times of
the Vedas, the ritual literature which gave directions for constructing
sacrificial fires at different times of the year dealt with the their
measurement and construction in a systematic and logical way, thus giving rise
to the *Sulba Sutras*. The construction
of altars (*vedi*) and the location of
sacrificial fires had to conform to clearly laid down instructions about their
shapes and areas in order that they may
be effective instruments of sacrifice. The *Sulba
Sutras* provide such instructions for two types of ritual - one for worship
at home and one for communal worship.
The instructions were mainly for the benefit of craftsmen laying out and
building the altars. Bodhayana,
Apastamba and Katyayana who have
recorded these *Sulbasutras* were not
only priests in the conventional sense but must have been craftsmen
themselves. The earliest of them, The *Bodhayana Sutras* , in three chapters,
(800 - 600 B.C.) contains a general
statement of the Pythagorean theorem, an approximation procedure for obtaining
the square root of two correct to five decimal places and a number of geometric
constructions. These latter include an
approximate squaring the circle, and construction of rectilinear shapes whose
area is equal to the sum or difference of areas of other shapes. The Bodhayana version of the Pythagorean
theorem sates as follows:

*The rope which is stretched across the
diagonal of a square *

*produces an area double the size of the
original square*.

It is therefore in the
fitness of things that the Pythagorean theorem of Mathematics may be renamed as
the Bodhayana theorem.! The other *sutras*
are two centuries later but all of them are prior to Panini of the fourth
century B.C. The geometry arising from
these *sutras* give several geometric
constructions. Some of these are:

1.
To merge two equal or
unequal squares to obtain a third square.

2.
To transform a
rectangle into a square of equal area

3.
Squaring a circle and
circling a square (approximately)

A remarkable achievement was
the discovery of a procedure for evaluating square roots to a high degree of
approximation. The square root of two is obtained as

1.4142156 …

the true value being
1.414213… . The fact that such procedures were used
successfully by the *Sulbasutra*
geometers to operations with other irrational numbers, is clear proof for
negating the western-held opinion that the *Sulba
sutra* geometers borrowed their methods from the Babylonians. The latter's
calculation of the square root of two is an isolated instance and further they
used the sexagesimal notation for numbers. The achievement of geometrical
constructs in Indian mathematics reached its peak later when they arrived at
the construction of *Sriyantra*, which is a complicated diagram, consisting of
nine interwoven isosceles triangles, four pointing upwards and four pointing
downwards. The triangles are arranged in such a way that they produce 43
subsidiary triangles, at the centre of
the smallest of which there is a big dot called the *bindu*. The difficult problem
is to construct the diagram in such a way that all the intersections are
correct and the vertices of the largest
triangles fall on the circumference of the enclosing circle. In all cases the base angles of the largest
triangles is about 51.5 degrees. This
has connections with the two most famous irrational numbers of Mathematics,
namely p and f. The quantity f, called the golden
ratio, has remarkable mathematical
properties and is almost a semi-mystical number.

**Section 4. Jaina contribution to
Fundamentals of Numbers**

By the time of the Jains,
the role of rituals in the development of mathematics declined
and mathematics began to be pursued also for its own sake. The Jains had a fascination for large
numbers. Their definitions of the various types of infinities they
comprehended are sophisticated, though
lacking in mathematical precision. But
it must be said to their credit that they were the first, in the chronology of
scientific thinking, to have recognised
that all infinities were not the same or equal. In fact this idea was established in the
mathematical world only in the latter half of the nineteenth century when
Cantor initiated his theory of sets.

The Jains were also aware of
the theory of indices, though they did not have the modern notation or any
convenient notation for the same. Calling the successive squares and square
roots as the first, the second, etc. they make the following statement: *The first square root multiplied by the
second square root is the cube of the second square root.* In modern
notation this is nothing but the identity in the theory of indices:

a^{1/2} x a^{1/4} = (a^{1/4
})^{3}

They have several such rules
for working with powers of a number. They also seem to have had an idea of the
logarithm of a number though they don't seem to have put them to practical use
in calculation. Another favourite topic
with them was the study of permutations and combinations. They had also a great
interest in sequences and progressions developed out of their philosophical
theory of cosmological structures. A
Jain canonical text entitled *Triloka
prajnApati* has a very detailed treatment of arithmetic progressions.

**Section 5: The Anonymous Bakshali Manuscript**

This manuscript was
discovered in 1881 A.D. near a village called Bakshali. It is written in an old
form of Sanskrit on seventy leaves of birch bark. It is probably a copy of a manuscript
composed in the early centuries A.D. It
is a handbook of rules and illustrative examples together with solutions, all
mainly on arithmetic and algebra. Fractions,
Square roots, Profit and Loss, Interest, Rule of Three, Approximation to surds,
Simple equations as well as Simultaneous equations, Quadratic equations,
Arithmetic and Geometric Progressions --
all these are covered. Very
unusually in the entire history
of Ancient Indian mathematics, the subject matter is organised in a
sequence: first, a rule or a sutra; then a relevant example in word form; the
same in notational form; then the solution and finally the demonstration or the
proof. Here for the first time in the
history of world mathematics, the Rule of Three is stated in its abstract form.
It was from here that the rule was taken to

If p yields f what will i yield?

Here p stands for *pramAna*,
f for *phala *and i for *icchA*.

Here p and i are of the same denomination and

f is of a different denomination.

Write p, f, i in that order. Multiply the middle quantity by
the last quantity and divide by the first.
The result is fi / p.

The first appearance of
indeterminate equations is in the
Bakshali mss. This marks the beginning of the continued work on indeterminate
equations in

**Section 6. Astronomy**

The contribution to
Astronomy by ancient Indians is so great that it does not befit it to include
it as one of the contributions of Indian Mathematics to the rest of the world.
It needs a separate forum all for itself. We shall leave it right there except
to add a note on the ancient contribution to the problem of telling time at
night by a look at the stars on the meridian. This part is usually not
emphasized.

The ancients of

*krittikA
simhe kAyA*

says that if you see the
asterism *krittikA *(Pleides, in modern
terminology) on the meridian, that is the time the Leo (= *simha*) constellation (of the zodiac) has risen above the horizon by
an amount indicated by the word: *kAyA*.
This latter word interpreted in *katapayA
sankhyA*, which is the notation used by astronomers, astrologers and
mathematicians to represent numbers, means in this context that the amount of
Leo above the (Eastern) horizon is 27 minutes of time. From this and the known
position of the Sun on the date in question, one mentally calculates the time
of night. On November 7 for example, the Sun is in the middle of Scorpio. So if
you see *krittikA *on the meridian it
means Leo has risen 27 minutes before and this means the Sun is behind by

93^{m} (remaining portion of *simha*)

+ 2^{h} (full
portion of *kanyA*)

+ 2^{h} (full portion of *tulA*

+ 60^{m} (half
portion of *vrischika*)

that is 6 hours 33 minutes. In other
words it is 6^{h} 33^{m} before sunrise. So it is 11-27 P.M.

Suffice it to say these
beautiful formulae constitute an intellectual marvel put to the most mundane
use. Never perhaps was so much achieved with so little so early in the Ascent
of Man. For details on this, see reference no. [10].

**Section 7.**
**Classical contribution to Indeterminate
Equations and Algebra**

The apex of Mathematical achievement of ancient

Aryabhata wrote the famous Aryabhatiyam which is an
exhaustive exposition of Astronomy. In
addition he gave a unique method of representing large numbers by word forms .
He systematized all the knowledge of astronomy and mathematics prior to him.
The first one in Indian mathematics to give the formula for the area of a
triangle was Aryabhata. Several results on Triangles and circles and on Progressions,
algorithm for finding cube roots, approximation of p, all these give him a unique position in the
development of mathematics. Aryabhata ushered in a Renaissance in Indian
Mathematics and Astronomy, that resulted in a remarkable flourishing of science
and technology in

Bhaskara I takes a large share of the credit of explaining the too brief and
aphoristic statements of Aryabhata. On the
important topic of indeterminate equations
the *Kuttaka* method was
introduced by Aryabhata and elucidated
by Bhaskara I.

Brahmagupta is generally known as the Indian
mathematician par excellence. His monumental work *Brahma SiddhAnta* has 24 chapters of which the latter 14 contain original results on arithmetic
algebra and on astronomical instruments.
The 12^{th} chapter is on mensuration. The 18^{th}
chapter is on *Kuttaka*. Among his famous results are those on rational
right-angled triangles, and cyclic quadrilaterals. He is the earliest one, in the history of world
mathematics, to have discussed cyclic
quadrilaterals. There is every reason for us to name cyclic quadrilaterals as *Brahmagupta Quadrilaterals*. It was partly through a translation of *Brahma-siddhAnt*a that the Arabs became
aware of Indian astronomy and mathematics.

Bhaskara II's famous work *SiddhAnta Siromani* has four parts of which the first two are
Mathematics and the latter two are astronomy. The first part, *LilAvati* is an extremely popular text
dealing with arithmetic, algebra, geometry and mensuration. The second part, *BIjaganitam* is a treatise on Advanced
Algebra. It contains problems on
determining unknown quantities, evaluating surds and solving simple and
quadratic equations.

The sheer ingenuity and versatility of Brahmagupta's
approach to indeterminate equations of the second degree of the form

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

is the climax of Indian work in this area. Bhaskara
II's *cakravAla*
method to solve such equations is world-famous. By using this powerful method
he solved, as one example, the above equation
with N = 61 and gave the least integral solution as

x = 226153980 and y =
1766319049.

The famous French mathematician, Fermat, in 1657
A.D. proposed this equation with N = 61 for solution as a challenge to his
contemporaries. None of them succeeded in solving the equation in integers. It
was not until 1767 A.D. that the western world through Euler, by Lagrange's
method of continued fractions, had a complete solution to such types of equations,
wrongly called Pell's equation by Euler. But the very same equation, though
coincidentally, was completely solved by Bhaskara II five hundred years
earlier.

The problem of determining integer solutions of such equations is called Diophantine Analysis
after the Greek Mathematician Diophantus (3^{rd} cen. A.D.). As soon as
one finds a non-trivial solution (that
is, other than the obvious solution x = 0, y = 1) an infinite number of new solutions can be
found by repeated application of the Principle of Compositions, known as Brahmagupta's Bhavana Principle. It is Bhaskara's *cakravAla* method that makes the decisive step in determining a
non-trivial solution. Under these
circumstances it is appropriate to designate these equations as the *Brahmagupta-Bhaskara equations*.

Before we leave this topic it is important to
mention Srinivasa Ramanujan, the 20^{th} century genius, who revelled
in such problems - namely, to determine the possible cases in which a number
can be broken up into two or more equal sums of like or unlike powers or more generally to solve intermediate
problems in rational numbers.

Bhaskara II introduces also the notion of
instantaneous motion of planets. He clearly distinguishes between *sthUla gati* (average velocity) and *sUkshma gati* (accurate velocity) in
terms of differentials. He also gave
formulae for the surface area of a sphere and its volume, and volume of the
frustum of a pyramid. Suffice it to say that his work on fundamental
operations, his rules of three, five, seven, nine and eleven, his work on
permutations and combinations and his handling of zero all speak of a maturity,
a culmination of five hundred years of mathematical progress.

**Section 8. Indian Trigonometry**

Though Trigonometry goes
back to the Greek period, the character of the subject started to resemble
modern form only after the time of Aryabhata. From here it went to *jya*, *kojya*
and *ukramajya*. The first one is *r sin**a** *where *r* is the radius of the circle and *a* is the angle subtended at the centre. The second one is *r cos**a* and the third one is *r (1 -
cos**a**)*. By taking the radius
of the circle to be 1, we get the modern
trigonometric functions. Various relationships between the sine of an arc and
its integral and fractional multiples were used to construct sine tables for
different arcs lying between 0 and 90°.

**Section 9: Kerala contribution to Infinite
Series and Calculus**.

Kerala mathematicians
produced rules for second order interpolation to calculate intermediate sine
values. The Kerala mathematician Madhava may have discovered the sine and
cosine series about three hundred years before *Aryabhatiya
bhashya* and *tantra-sangraha*
contain work on infinite-series expansions, problems of algebra and spherical
geometry.

**Section 10. Modern Contribution: **

**Srinivasa Ramanujan onwards**

The second decade
of the 20^{th} century compulsorily turned the attention of the
mathematical world to ^{th}
century saw the growth of Mathematics in

Copyright © V. Krishnamurthy January 2007

***********

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Copyright © V.
Krishnamurthy January, 2007