This was published in National newspaper THE HINDU ....26 th december 2011 in science and technology section ....
It is high time we studied our mathematical heritage with diligence and objectivity.
Quite
often I find that conversations, with people from various walks of
life, on ancient Indian mathematics slide to “Vedic mathematics” of the
“16 sutras” fame, which is supposed to endow one with magical powers of
calculation. Actually, the “16 sutras” were introduced by Bharati
Krishna Tirthaji, who was the Sankaracharya of Puri from 1925 until he
passed away in 1960, associating with them procedures for certain
arithmetical or algebraic computations. Thus, this so-called “Vedic
mathematics (VM)” is essentially a 20th century phenomenon.
Neither
the “sutras” nor the procedures that they are supposed to yield, or
correspond to, have anything to do with either the Vedas, or even with
any post-Vedic mathematical tradition of yore in India. The image that
it may conjure up of ancient rishis engaged in such arithmetical
exercises as are taught to the children in the name of VM, and
representing the solutions through word-strings of a few words in modern
styled Sanskrit, with hardly any sentence structure or grammar, is just
too far from the realm of the plausible. It would have amounted to a
joke, but for the aura it has acquired on account of various factors,
including the general ignorance about the knowledge in ancient times. It
is a pity that a long tradition of over 3,000 years of learning and
pursuit of mathematical ideas has come to be perceived by a large
section of the populace through the prism of something so mundane and so
lacking in substance from a mathematical point of view, apart from not
being genuine.
Tall claims
The
colossal neglect involved is not for want of pride about the
achievements of our ancients; on the contrary, there is a lot of writing
on the topic, popular as well as technical, that is full of
unsubstantiated claims conveying an almost supreme knowledge our
forefathers are supposed to have possessed. But there is very little
understanding or appreciation, on an intellectual plane, of the specics
of their knowledge or achievements in real terms.
In
the colonial era this variety of discourse emerged as an antithesis to
the bias that was manifest in the works of some Western scholars. Due to
the urgency to respond to the adverse propaganda on the one hand and
the lack of resources in addressing the issues at a more profound level
on the other, recourse was often taken to short-cuts, which involved
more assertiveness than substance. There were indeed some Indian
scholars, like Sudhakar Dvivedi, who adhered to a more intellectual
approach, but they were a minority. Unfortunately, the old discourse has
continued long after the colonial context is well past, and long after
the world community has begun to view the Indian achievements with
considerable objective curiosity and interest. It is high time that we
switch to a mode betting a sovereign and intellectually self-reliant
society, focussing on an objective study and critical assessment,
without the reference frame of “what they say” and how “we must assert
ourselves.”
Ancient
India has indeed contributed a great deal to the world's mathematical
heritage. The country also witnessed steady mathematical developments
over most part of the last 3,000 years, throwing up many interesting
mathematical ideas well ahead of their appearance elsewhere in the
world, though at times they lagged behind, especially in the recent
centuries. Here are some episodes from the fascinating story that forms a
rich fabric of the sustained intellectual endeavour.
Vedic knowledge
The
mathematical tradition in India goes back at least to the Vedas. For
compositions with a broad scope covering all aspects of life, spiritual
as well as secular, the Vedas show a great fascination for large
numbers. As the transmission of the knowledge was oral, the numbers were
not written, but expressed as combinations of powers of 10. It would be
reasonable to believe that when the decimal place value system for
written numbers came into being it owed a great deal to the way numbers
were discussed in the older compositions.
The
decimal place value system of writing numbers, together with the use of
‘0,' is known to have blossomed in India in the early centuries AD, and
spread to the West through the intermediacy of the Persians and the
Arabs. There were actually precursors to the system, and various
components of it are found in other ancient cultures such as the
Babylonian, Chinese, and Mayan. From the decimal representation of the
natural numbers, the system was to evolve further into the form that is
now commonplace and crucial in various walks of life, with decimal
fractions becoming part of the number system in 16th century Europe,
though this again has some intermediate history involving the Arabs. The
evolution of the number system represents a major phase in the
development of mathematical ideas, and arguably contributed greatly to
the overall advance of science and technology. The cumulative history of
the number system holds a lesson that progress of ideas is an inclusive
phenomenon, and while contributing to the process should be a matter of
joy and pride to those with allegiance to the respective contributors,
the role of others also ought to be appreciated.
It
is well-known that Geometry was pursued in India in the context of
construction of vedis for the yajnas of the Vedic period. The
Sulvasutras contain elaborate descriptions of construction of vedis and
enunciate various geometric principles. These were composed in the rst
millennium BC, the earliest Baudhayana Sulvasutra dating back to about
800 BC. Sulvasutra geometry did not go very far in comparison to the
Euclidean geometry developed by the Greeks, who appeared on the scene a
little later, in the seventh century BC. It was, however, an important
stage of development in India too. The Sulvasutra geometers were aware,
among other things, of what is now called the Pythagoras theorem, over
200 years before Pythagoras (all the four major Sulvasutras contain an
explicit statement of the theorem), addressed (within the framework of
their geometry) issues such as nding a circle with the same area as a
square and vice versa, and worked out a very good approximation to the
square root of two, in the course of their studies.
Though
it is generally not recognised, the Sulvasutra geometry was itself
evolving. This is seen, in particular, from the differences in the
contents of the four major extant Sulvasutras. Certain revisions are
especially striking. For instance, in the early Sulvasutra period the
ratio of the circumference to the diameter was, as in other ancient
cultures, thought to be three, as seen in a sutra of Baudhayana, but in
the Manava Sulvasutra, a new value was proposed, as three-and-one-fth.
Interestingly, the sutra describing it ends with an exultation “not a
hair-breadth remains,” and though we see that it is still substantially
off the mark, it is a gratifying instance of an advance made. In the
Manava Sulvasutra one also nds an improvement over the method described
by Baudhayana for nding the circle with the same area as that of a given
square.
The
Jain tradition has also been very important in the development of
mathematics in the country. Unlike for the Vedic people, for Jain
scholars the motivation for mathematics came not from ritual practices,
which indeed were anathema to them, but from the contemplation of the
cosmos. Jains had an elaborate cosmography in which mathematics played
an integral role, and even largely philosophical Jain works are seen to
incorporate mathematical discussions. Notable among the topics in the
early Jain works, from about the fifth century BC to the second century
AD, one may mention geometry of the circle, arithmetic of numbers with
large powers of 10, permutations and combinations, and categorisations
of innities (whose plurality had been recognised).
As
in the Sulvasutra tradition, the Jains also recognised, around the
middle of the rst millennium BC, that the ratio of the circumference of
the circle to its diameter is not three. In “Suryaprajnapti,” a Jain
text believed to be from the fourth century BC, after recalling the
“traditional” value three for it, the author discards that in favour the
square root of 10. This value for the ratio, which is reasonably close
to the actual value, was prevalent in India over a long period and is
often referred as the Jain value. It continued to be used long after
Aryabhata introduced the well-known value 3.1416 for the ratio. The Jain
texts also contain rather unique formulae for lengths of circular arcs
in terms of the length of the corresponding chord and the bow (height)
over the chord, and also for the area of regions subtended by circular
arcs together with their chords. The means for the accurate
determination of these quantities became available only after the advent
of Calculus. How the ancient Jain scholars arrived at these formulae,
which are close approximations, remains to be understood.
Jain tradition
After
a lull of a few centuries in the early part of the rst millennium,
pronounced mathematical activity is seen again in the Jain tradition
from the 8th century until the middle of the 14th century.
Ganitasarasangraha of Mahavira, written in 850, is one of the well-known
and inuential works. Virasena (8th century), Sridhara (between 850 and
950), Nemicandra (around 980 CE), Thakkura Pheru (14th century) are some
more names that may be mentioned. By the 13th and 14th centuries,
Islamic architecture had taken root in India and in Ganitasarakaumudi of
Thakkura Pheru, who served as treasurer in the court of the Khilji
Sultans in Delhi, one sees a combination of the native Jain tradition
with Indo-Persian literature, including work on the calculation of areas
and volumes involved in the construction of domes, arches, and tents
used for residential purposes.
Mathematical
astronomy or the Siddhanta tradition has been the dominant and enduring
mathematical tradition in India. It ourished almost continuously for
over seven centuries, starting with Aryabhata (476-550) who is regarded
as the founder of scientic astronomy in India, and extending to Bhaskara
II (1114-1185) and beyond. The essential continuity of the tradition
can be seen from the long list of prominent names that follow Aryabhata,
spread over centuries: Varahamihira in the sixth century, Bhaskara I
and Brahmagupta in the seventh century, Govindaswami and Sankaranarayana
in the ninth century, Aryabhata II and Vijayanandi in the 10th century,
Sripati in the 11th century, Brahmadeva and Bhaskara II in the 12th
century, and Narayana Pandit and Ganesa from the 14th and 16th centuries
respectively.
Aryabhatiya,
written in 499, is basic to the tradition, and even to the later works
of the Kerala school of Madhava (more on that later). It consists of 121
verses divided into four chapters — Gitikapada, Ganitapada,
Kalakriyapada and Golapada. The rst, which sets out the cosmology,
contains also a verse describing a table of 24 sine differences at
intervals of 225 minutes of arc. The second chapter, as the name
suggests, is devoted to mathematics per se, and includes in particular
procedures to nd square roots and cube roots, an approximate expression
for ‘pi' (amounting to 3.1416 and specied to be approximate), formulae
for areas and volumes of various geometric gures, and shadows, formulae
for sums of consecutive integers, sums of squares, sums of cubes and
computation of interest. The other two chapters are concerned with
astronomy, dealing with distances and relative motions of planets,
eclipses and so on.
Influential work
Brahmagupta's
Brahmasphutasiddhanta is a voluminous work, especially for its time, on
Siddhanta astronomy, in which there are two chapters, Chapter 12 and
Chapter 18, devoted to general mathematics. Incidentally, Chapter 11 is a
critique on earlier works including Aryabhatiya; as in other healthy
scientific communities this tradition also had many, and often bitter,
controversies. Chapter 12 is well-known for its systematic treatment of
arithmetic operations, including with negative numbers; the notion of
negative numbers had eluded Europe until the middle of the second
millennium. The chapter also contains geometry, including in particular
his famous formula for the area of a quadrilateral (stated without the
condition of cyclicity of the quadrilateral that is needed for its
validity — a point criticised by later mathematicians in the tradition).
Chapter 18 is devoted to the kuttaka and other methods, including for
solving second-degree indeterminate equations. An identity described in
the work features also in some current studies where it is referred as
the Brahmagupta identity. Apart from this, Chapter 21 has verses dealing
with trigonometry. Brahmasphutasiddhanta considerably influenced
mathematics in the Arab world, and in turn the later developments in
Europe. Bhaskara II is the author of the famous mathematical texts
Lilavati and Bijaganita. Apart from being an accomplished mathematician
he was a great teacher and populariser of mathematics. Lilavati, which
literally means ‘one who is playful,' presents mathematics in a playful
way, with several verses directly addressing a pretty young woman, and
examples presented through reference to various animals, trees,
ornaments, and so on. (Legend has it that the book is named after his
daughter after her wedding failed to materialise on account of an
accident with the clock, but there is no historical evidence to that
effect.) The book presents, apart from various introductory aspects of
arithmetic, geometry of triangles and quadrilaterals, examples of
applications of the Pythagoras theorem, trirasika, kuttaka methods,
problems on permutations and combinations, etc. The Bijaganita is an
advanced-level treatise on Algebra, the first independent work of its
kind in Indian tradition. Operations with unknowns, kuttaka and
chakravala methods for solutions of indeterminate equations are some of
the topics discussed, together with examples. Bhaskara's work on
astronomy, Siddhantasiromani and Karana kutuhala, contain several
important results in trigonometry, and also some ideas of Calculus.
The
works in the Siddhanta tradition have been edited on a substantial
scale and there are various commentaries available, including many from
the earlier centuries, and works by European authors such as Colebrook,
and many Indian authors including Sudhakara Dvivedi, Kuppanna Sastri and
K.V. Sarma. The two-volume book of Datta and Singh and the book of
Saraswati Amma serve as convenient references for many results known in
this tradition. Various details have been described, with a
comprehensive discussion, in the recent book by Kim Plofker. The
Bakhshali manuscript, which consists of 70 folios of bhurjapatra (birch
bark), is another work of signicance in the study of ancient Indian
mathematics, with many open issues around it. The manuscript was found
buried in a eld near Peshawar, by a farmer, in 1881. It was acquired by
the Indologist A.F.R. Hoernle, who studied it and published a short
account on it. He later presented the manuscript to the Bodleian Library
at Oxford, where it has been since then. Facsimile copies of all the
folios were brought out by Kaye in 1927, which have since then been the
source material for the subsequent studies. The date of the manuscript
has been a subject of much controversy since the early years, with the
estimated dates ranging from the early centuries of CE to the 12th
century.
Takao
Hayashi, who produced what is perhaps the most authoritative account so
far, concludes that the manuscript may be assigned sometime between the
eighth century and the 12th century, while the mathematical work in it
may most probably be from the seventh century. Carbon dating of the
manuscript could settle the issue, but efforts towards this have not
materialised so far.
A
formula for extraction of square-roots of non-square numbers found in
the manuscript has attracted much attention. Another interesting feature
of the Bakhshali manuscript is that it involves calculations with large
numbers (in decimal representation).
Kerala school
Let
me nally come to what is called the Kerala School. In the 1830s,
Charles Whish, an English civil servant in the Madras establishment of
the East India Company, brought to light a collection of manuscripts
from a mathematical school that ourished in the north-central part of
Kerala, between what are now Kozhikode and Kochi. The school, with a
long teacher-student lineage, lasted for over 200 years from the late
14th century well into the 17th century. It is seen to have originated
with Madhava, who has been attributed by his successors many results
presented in their texts. Apart from Madhava, Nilakantha Somayaji was
another leading personality from the school. There are no extant works
of Madhava on mathematics (though some works on astronomy are known).
Nilakantha authored a book called Tantrasangraha (in Sanskrit) in 1500
AD. There have also been expositions and commentaries by many other
exponents from the school, notable among them being Yuktidipika and
Kriyakramakari by Sankara, and Ganitayuktibhasha by Jyeshthadeva which
is in Malayalam. Since the middle of the 20th century, various Indian
scholars have researched on these manuscripts and the contents of most
of the manuscripts have been looked into. An edited translation of the
latter was produced by K.V. Sarma and it has recently been published
with explanatory notes by K. Ramasubramanian, M.D. Srinivas and M.S.
Sriram. An edited translation of Tantrasangraha has been brought out
more recently by K. Ramasubramanian and M.S. Sriram.
The
Kerala works contain mathematics at a considerably advanced level than
earlier works from anywhere in the world. They include a series
expansion for ‘pi' and the arc-tangent series, and the series for sine
and cosine functions that were obtained in Europe by Gregory, Leibnitz
and Newton, respectively, over 200 years later. Some numerical values
for ‘pi' that are accurate to 11 decimals are a highlight of the work.
In many ways, the work of the Kerala mathematicians anticipated calculus
as it developed in Europe later, and in particular involves
manipulations with indenitely small quantities (in the determination of
circumference of the circle and so on) analogous to the innitesimals in
calculus; it has also been argued by some authors that the work is
indeed calculus already.
Vedic maths is based on sixteen sutra's or principles . These principles are general in nature and can be applied in many ways . In practice, the vedic system is used to solve difficult problems or huge sums in more effective and easy way.These method are just a part of a complete system of mathematics which is far more systematic than the modern system taught now. The simplicity of vedic mathematics helps us to solve the calculations mentally with sifficient practice.
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