History of Mathematics -
The Indian Contribution
and Sanskrit the mother of Europe's languages.
India was the mother of our philosophy,
of much of our mathematics, of the ideals embodied in
Christianity... of self-government and democracy.
In many ways, Mother India is the mother of us all."
- Will Durant
- American Historian 1885-1981
The Contribution to Mathematics by India
can be divided into ten categories:
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 3rd 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 South
America had a zero in the first century A.D. . but they did not use it in a fixed base
system. The Greeks were hampered by their use of letters for the numbers. Before zero was invented, the art of
reckoning remained an exclusive and
highly skilled profession. It was difficult to distinguish, say, 27,
207, 270, 2007, because the latter three were all written 2 7, with a ‘space’
in between. The positional system is not possible in the Roman numeral system which had no expression or symbol for
zero. A number, say, 101,000, would have to be written only by 101 consecutive
M’s. The Egyptians had no zero and never reached the idea of expressing all
numbers with ten digits. The mathematical climate among the Hindus, however,
was congenial for the invention of zero and for its use as the null-value in
all facets of calculation, due to
four factors:
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 Spain
around 700 A.D. . Later, when massive
Latin translations of books from Baghdad took place around the close of the first
millenium A.D. the concept was found in
an arithmetic book dated 820 A.D. . by Muhammad Ibn Musa al-Khouarizmi who
explained the whole system in great detail. The Arabs themselves had no number
system of their own. It was the Hindu system he was explaining. That is why the numerals are called the
Indo-Arabic numerals even today. It is however a misnomer to call them so;
they are already found on the Rock Edits of Ashoka (256 B.C. .). In spite of its being so crucial to our
living, it took centuries for the western world to appreciate and incorporate
this most valuable numeral, zero, in their recording of accounts or in
scholarly writings. For by the time ‘Zero’ reached the West, the Dark Ages of
the western world had begun. There are traces, however, of its knowledge in Spain in the tenth century A.D. .
But the final breakthrough of the introduction to the West was by Leonardo of Pisa, through his popular
text Liber Abaci, 1202 A.D. ., The first European book (in French) that used the zero appeared in 1275.
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 Europe some countries banned the
positional number system, along with zero, brought by the Arabs whom they considered as
heathens. So they took 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:
a1/2 x a1/4 = (a1/4
)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 Europe
by the Arabs and it was then known as the Golden Rule. It became very popular
in Europe after the Renaissance. The rule goes
as follows:
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 India.
Such an interest, though originally generated from the demands made by
astronomical calculations, was also pursued for its own sake, sometimes even
for recreational purposes. The ease with which the place value system is worked
in the Bakshali mss. suggests that the system predates the mss. by probably a
few centuries.
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 India have passed on to us 27
mathematical formulae coded in the Sanskrit language, but not very difficult to
remember. In fact, very possibly it has mostly come down to us by oral
transmission from generation to generation.
For instance, the formula
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
93m (remaining portion of simha)
+ 2h (full
portion of kanyA)
+ 2h (full portion of tulA)
+ 60m (half
portion of vrischika)
that is 6 hours 33 minutes. In other
words it is 6h 33m 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 India occurred during the so-called classical
period of Indian Mathematics. The great names are: Aryabhata I (b.476 A.D.) ;
Brahmagupta (b.598 A.D.); Bhaskara I (circa 620 A.D.) ; Mahavira (circa 850
A.D.); Sridhara (circa 900 A.D.) ; Bhaskara II (b.1114 A.D.); Nilakantha
Somayaji (1445 - 1545 A.D.).
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 India. Aryabhata, for the first time, secularised
mathematics and astronomy in India
and established these as intellectual disciplines in their own right. Excellent commentators followed
Aryabhata and to them are due several
modifications and applications, explanation of subtle points, and finally, the
proofs of results embodied in the sutras
of the Master.
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 12th chapter is on mensuration. The 18th
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-siddhAnta 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 x2 + 1 = y2
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 (3rd 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 20th 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 Europe through the Arabs and went into several
modifications to reach its present form.
In ancient times Trigonometry was considered a part of astronomy. Three functions were introduced: jya, kojya
and ukramajya. The first one is r sina where r is the radius of the circle and a is the angle subtended at the centre. The second one is r cosa and the third one is r (1 -
cosa). 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 Newton.
In this sense we may consider Madhava to have been the founder of
mathematical analysis. Madhava (circa 1340 - 1425 A.D. ) was the first to take
the decisive step from the finite procedures of ancient Indian mathematics to
treat their limit-passage to infinity.
His contributions include infinite-series expansions of circular and
trigonometric functions and finite-series approximations. His power series for p and for sine and cosine
functions is referred to reverentially
by later writers. Many later discoveries in European mathematics ( for example,
the Gregory series for the inverse tangent)
were anticipated by Kerala astronomer-mathematicians. Nilakantha was mainly an astronomer but his Aryabhatiya
bhashya and tantra-sangraha
contain work on infinite-series expansions, problems of algebra and spherical
geometry.
Section 10. Modern Contribution:
Srinivasa Ramanujan onwards
Mathematics represents a high level of abstraction attained by the human mind. In India, mathematics has its roots in Vedic literature which is nearly 4000 years old. Between 1000 B.C. and 1000 A.D. various treatises on mathematics were authored by Indian mathematicians in which were set forth for the first time, the concept of zero, the techniques of algebra and algorithm, square root and cube root.
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This method of graduated calculation was documented in the Pancha-Siddhantika (Five Principles) in the 5th Century But the technique is said to be dating from Vedic times circa 2000 B.C. |
Even the technique of calculation, called algorithm, which is today widely used in designing soft ware programs (instructions) for computers was also derived from Indian mathematics. In this chapter we shall examine the advances made by Indian mathematicians in ancient times.
ALGEBRA- THE OTHER MATHEMATICS ?
In India around the 5th century A.D. a sys tem of mathematics that made astronomical calculations easy was developed. In those times its application was limited to astronomy as its pioneers were Astronomers. As tronomical calculations are complex and involve many variables that go into the derivation of unknown quantities. Algebra is a short-hand method of calculation and by this feature it scores over conventional arithmetic.
In ancient India conventional mathematics termed Ganitam was known before the development of algebra. This is borne out by the name - Bijaganitam, which was given to the algebraic form of computation. Bijaganitam means 'the other mathematics' (Bija means 'another' or 'second' and Ganitam means mathematics). The fact that this name was chosen for this system of computation implies that it was recognised as a parallel system of computation, different from the conventional one which was used since the past and was till then the only one. Some have interpreted the term Bija to mean seed, symbolizing origin or beginning. And the inference that Bijaganitam was the original form of computation is derived. Credence is lent to this view by the existence of mathematics in the Vedic literature which was also shorthand method of computation. But whatever the origin of algebra, it is certain that this technique of computation Originated in India and was current around 1500 years back. Aryabhatta an Indian mathematican who lived in the 5th century A.D. has referred to Bijaganitam in his treatise on Mathematics, Aryabhattiya. An Indian mathematician - astronomer, Bhaskaracharya has also authored a treatise on this subject. the treatise which is dated around the 12th century A.D. is entitled 'Siddhanta-Shiromani' of which one section is entitled Bijaganitam.
Thus the technique of algebraic computation was known and was developed in India in earlier times. From the 13th century onwards, India was subject to invasions from the Arabs and other Islamised communities like the Turks and Afghans. Alongwith these invader: came chroniclers and critics like Al-beruni who studied Indian society and polity.
The Indian system of mathematics could no have escaped their attention. It was also the age of the Islamic Renaissance and the Arabs generally improved upon the arts and sciences that they imbibed from the land they overran during their great Jehad. Th system of mathematics they observed in India was adapted by them and given the name 'Al-Jabr' meaning 'the reunion of broken parts'. 'Al' means 'The' & 'Jabr' mean 'reunion'. This name given by the Arabs indicates that they took it from an external source and amalgamated it with their concepts about mathematics.
Between the 10th to 13th centuries, the Christian kingdoms of Europe made numerous attempts to reconquer the birthplace of Jesus Christ from its Mohammedan-Arab rulers. These attempts called the Crusades failed in their military objective, but the contacts they created between oriental and occidental nations resulted in a massive exchange of ideas. The technique of algebr could have passed on to the west at thi time.
During the Renaissance in Europe, followed by the industrial revolution, the knowledge received from the east was further developed. Algebra as we know it today has lost any characteristics that betray it eastern origin save the fact that the tern 'algebra' is a corruption of the term 'Al jabr' which the Arabs gave to Bijaganitam Incidentally the term Bijaganit is still use in India to refer to this subject.
In the year 1816, an Englishman by the name James Taylor translated Bhaskara's Leelavati into English. A second English translation appeared in the following year (1817) by the English astronomer Henry Thomas Colebruke. Thus the works of this Indian mathematician astronomer were made known to the western world nearly 700 years after he had penned them, although his ideas had already reached the west through the Arabs many centuries earlier.
In the words of the Australian Indologist A.L. Basham (A.L. Basham; The Wonder That was India.) "... the world owes most to India in the realm of mathematics, which was developed in the Gupta period to a stage more advanced than that reached by any other nation of antiquity. The success of Indian mathematics was mainly due to the fact that Indians had a clear conception of the abstract number as distinct from the numerical quantity of objects or spatial extension."
Thus Indians could take their mathematical concepts to an abstract plane and with the aid of a simple numerical notation devise a rudimentary algebra as against the Greeks or the ancient Egyptians who due to their concern with the immediate measurement of physical objects remained confined to Mensuration and Geometry.
GEOMETRY AND ALGORITHM
But even in the area of Geometry, Indian mathematicians had their contribution. There was an area of mathematical applications called Rekha Ganita (Line Computation). The Sulva Sutras, which literally mean 'Rule of the Chord' give geometrical methods of constructing altars and temples. The temples layouts were called Mandalas. Some of important works in this field are by Apastamba, Baudhayana, Hiranyakesin, Manava, Varaha and Vadhula.
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The Buddhist Pagodas borrowed their plan of construction from the geometric grid of the Mandala used for constructing temples in India (A majestic Pagoda at Bangkok) |
The Arab scholar Mohammed Ibn Jubair al Battani studied Indian use of ratios from Retha Ganita and introduced them among the Arab scholars like Al Khwarazmi, Washiya and Abe Mashar who incorporated the newly acquired knowledge of algebra and other branches of Indian mathema into the Arab ideas about the subject.
The chief exponent of this Indo-Arab amalgam in mathematics was Al Khwarazmi who evolved a technique of calculation from Indian sources. This technique which was named by westerners after Al Khwarazmi as "Algorismi" gave us the modern term Algorithm, which is used in computer software.
Algorithm which is a process of calculation based on decimal notation numbers. This method was deduced by Khwarazmi from the Indian techniques geometric computation which he had st ied. Al Khwarazmi's work was translated into Latin under the title "De Numero Indico" which means 'of Indian Numerals' thus betraying its Indian origin. This translation which belong to the 12th century A.D credited to one Adelard who lived in a town called Bath in Britian.
Thus Al Khwarazmi and Adelard could looked upon as pioneers who transmit Indian numerals to the west. Incidents according to the Oxford Dictionary, word algorithm which we use in the English language is a corruption of the name Khwarazmi which literally means '(a person) from Khawarizm', which was the name of the town where Al Khwarazmi lived. To day unfortunately', the original Indian texts that Al Khwarazmi studied arelost to us, only the translations are avail able .
The Arabs borrowed so much from India the field of mathematics that even the subject of mathematics in Arabic came to known as Hindsa which means 'from India and a mathematician or engineer in Arabic is called Muhandis which means 'an expert in Mathematics'. The word Muhandis possibly derived from the Arabic term mathematics viz. Hindsa.
The Concept of Zero
The concept of zero also originated inancient India. This concept may seem to be a very ordinary one and a claim to its discovery may be viewed as queer. But if one gives a hard thought to this concept it would be seen that zero is not just a numeral. Apart from being a numeral, it is also a concept, and a fundamental one at that. It is fundamental because, terms to identify visible or perceptible objects do not require much ingenuity.
But a concept and symbol that connotes nullity represents a qualitative advancement of the human capacity of abstraction. In absence of a concept of zero there could have been only positive numerals in computation, the inclusion of zero in mathematics opened up a new dimension of negative numerals and gave a cut off point and a standard in the measurability of qualities whose extremes are as yet unknown to human beings, such as temperature.
In ancient India this numeral was used in computation, it was indicated by a dot and was termed Pujyam. Even today we use this term for zero along with the more current term Shunyam meaning a blank. But queerly the term Pujyam also means holy. Param-Pujya is a prefix used in written communication with elders. In this case it means respected or esteemed. The reason why the term Pujya - meaning blank - came to be sanctified can only be guessed.
Indian philosophy has glorified concepts like the material world being an illusion Maya), the act of renouncing the material world (Tyaga) and the goal of merging into the void of eternity (Nirvana). Herein could lie the reason how the mathematical concept of zero got a philosophical connotation of reverence.
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In a queer way the concept of 'Zero' or Shunya is derived from the concept of a void. The concept of void existed in Hindu Philosophy hence the derivation of a symbol for it. The concept of Shunyata, influenced South-east asian culture through the Buddhist concept of Nirvana 'attaining salvation by merging into the void of eternity' (Ornate Entrance of a Buddhist temple in Laos) |
It is possible that like the technique of algebra; the concept of zero also reached the west through the Arabs. In ancient India the terms used to describe zero included Pujyam, Shunyam, Bindu the concept of a void or blank was termed as Shukla and Shubra. The Arabs refer to the zero as Siphra or Sifr from which we have the English terms Cipher or Cypher. In English the term Cipher connotes zero or any Arabic numeral. Thus it is evident that the term Cipher is derived from the Arabic Sifr which in turn is quite close to the Sanskrit term Shubra.
The ancient India astronomer Brahmagupta is credited with having put forth the concept of zero for the first time: Brahmagupta is said to have been born the year 598 A.D. at Bhillamala (today's Bhinmal ) in Gujarat, Western India. ] much is known about Brahmagupta's early life. We are told that his name as a mathematician was well established when K Vyaghramukha of the Chapa dyansty m him the court astronomer. Of his two treatises, Brahma-sputa siddhanta and Karanakhandakhadyaka, first is more famous. It was a corrected version of the old Astronomical text, Brahma siddhanta. It was in his Brahma-sphu siddhanta, for the first time ever had be formulated the rules of the operation zero, foreshadowing the decimal system numeration. With the integration of zero into the numerals it became possible to note higher numerals with limited charecters.
In the earlier Roman and Babylonian systems of numeration, a large number of chara acters were required to denote higher numerals. Thus enumeration and computation became unwieldy. For instance, as E the Roman system of numeration, the number thirty would have to be written as X: while as per the decimal system it would 30, further the number thirty three would be XXXIII as per the Roman system, would be 33 as per the decimal system. Thus it is clear how the introduction of the decimal system made possible the writing of numerals having a high value with limited characters. This also made computation easier.
Apart from developing the decimal system based on the incorporation of zero in enumeration, Brahmagupta also arrived at solutions for indeterminate equations of 1 type ax2+1=y2 and thus can be called the founder of higher branch of mathematics called numerical analysis. Brahmagupta's treatise Brahma-sputa-siddhanta was translated into Arabic under the title Sind Hind).
For several centuries this translation mained a standard text of reference in the Arab world. It was from this translation of an Indian text on Mathematics that the Arab mathematicians perfected the decimal system and gave the world its current system of enumeration which we call the Arab numerals, which are originally Indian numerals.
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