Kerala and Calculus
https://ramamohanraocheruku.blogspot.com/2021/11/kerala-and-calculus.html
According to Bhagavata Purana and the other scriptures the
civilisation and culture had started in Kerala lakhs of years ago. The
civilisation in Kerala is different from other parts of ‘Bharata’ is different
in many facets. The Martial Art ‘Kalaripayattu’. The Keralites honestly trust
that Parushurama is the founder of ‘Kalaripayattu’, Ayurvedam and Vedanga
Jyothishyam. The Brahmanic and Vedic culture of Kerala also varies from the
other parts of Bharat. Ayurvedic culture of Kerala is unique. People opted for
naval rout alone could reach Kerala but not those who crossed Hindukush.
‘Vishu’ ‘Onam’ are the grand festivals they celebrate without religion. Kerala
is the hub of all spices and Malabar cost in particular.
Saint Thomas among the 12 apostles of Chris said to have landed in
the Malabar ((May be meeting of water with the base of the row of mountains.) cost.
The first Indian mosque was built in Kalankallur (Kodungallur) by Muhammad ibn
Malik. Mappila Muslims are but one among the many communities that form the
first Muslim population of Kerala, as I presume. "Mappila' means
'Son-in-law'. Probably the intruders who got settled in Kerala by marrying
Kerala girls are termed as 'Mappila'. For a true Bharatiya it sounds pathetic.
Muslims are referred to as 'Yavanaka Mappila' and the Saint Thomous Christians
are called 'Nasrani Mappila' and Cochin Jews are called 'Juda Mappila'. However
the word 'Mappila' is heart rending t note that Kerala could not find any
appropriate word for the immigrants.
Kerala gave to the world ‘Jagadguru Sankaracharya’ who is unparallel
to any philosopher till date. Added to this Kerala gave many great
mathematicians, Astronomers, and Ayurvedic Bhishgvaras (Doctors).
Though this introduction is unwanted and unnecessary for 'Kerala and
Calculus'I took this as the opportunity to tell about the dominant religions
other than 'Sanathana Dharma'. Kerala, especially Malabar stood as the portal
for knowledge transfer. As an example the Syrian Catholics settled in Malabar
centuries before and the Catholic scholar and Arch Bishop comes back to India
and exports the knowledge of our Numerals to Western World.
Vaskodagama in fact did not discover India nor did he claim to have
discovered in his writings. It is all the European, left and Muslim writers who
distorted our History and the then Government cared a tuppence to the writings.
He was said to have been helped by a Gujarati ‘Trade Sailor’ having trade links
with South Africa especially Lisbon, who was provided by the then king. Pliny
and Ptolemy (105=170 AD) also mentioned that the sea route and trade were well
established with Bharat through ‘Kerala’. Muziris in the Malabar cost as
mentioned in the writing of the then foreigners, is still to be located with
its present name.
In the
year 662, Severus returns again to his lost subject (Fol. 170) and says that
the Greeks learnt Astronomy from the Chaldeans who are the Syrians, which might
have gone overseas through voyages from ‘Bharata’. He concludes that learning
belongs to all and gives as an example the Hindus who have found a means of
expressing all the numbers with 9 signs. This is here the oldest Eastern
Mention of the Indian figures. Which have been published in the journal
Asiatique, Oct. 1910, p. 225-7.
Kerala
has thousands of years of ancient Vedic Culture, rich heritage. We would have
studied at the High School Level about
Geometry:
Which is about their shapes and their properties
Algebra:
About proportions and their relationships.
Probability:
About chances and occurrences.
Calculus:
Calculating the rate of change of any given aspect.
Let
us elaborate this a bit. Suppose a cricket batter hits the ball, it will create
a path till touching the ground. What is the Velocity of the ball at its peak height
is the derivative and what the area is swept by the ball is the integration. These
are the two main components dealt in Calculus. Coming back, let us think about
when will the ball hit the ground in time? In a nutshell derivative or
differential calculus deals about calculating the implied properties of the
ball like position of the ball velocity of the ball and change in the velocity
etc. Integration deals with magnitudes like lengths areas and volumes.
Let
us just touch Khagola Sastra.
In
fact Jyotisha Sastra and Khagola sastra are two different subjects. While Khagola Sastra deals with the distance
of the planets, there volume and their rotation round the sun etc. Jyotisha
Sastra deals with the light received on earth by various planets and the
relative effect on mankind. Calculus deals with 1. Time Computation, 2. Rate of
change of planetary positions 3. Size and velocity of planets. In Fact
trigonometry and Geometry also have their organic roots in Khagola Sastra. The
root word ‘Trikonamithi’ is the origin for ‘Trigonometry’ and ‘Jyamithi’ is for
‘Geometry’, taken from Sanskrit.
Madhava of Sangamagrama (location with the modern name yet to be
known), the follower of Aryabhata school of Astronomy, has been called
"the greatest mathematician-astronomer of medieval India", or as
"the founder of mathematical analysis; some of his discoveries in this
field show him to have possessed extraordinary intuition". O'Connor and
Robertson state that a fair assessment of Madhava is that he took the decisive
step towards modern classical analysis.
Madhava laid the foundations for the development of calculus,
which were further developed by his successors at his Kerala school of
astronomy and mathematics (Certain ideas of calculus were known to earlier
mathematicians.) in 14th century. He also extended some results
found in earlier works, including those of Bhaskara II.
The Kerala School was well known in the 15th and 16th centuries,
by the period of the first contact with European navigators in the Malabar
Coast. At the time, the port of Muziris, near Sangamagrama, was a major center
for maritime trade, and a number of Jesuit missionaries and traders were active
in this region. Given the fame of the Kerala School, and the interest shown by
some of the Jesuit groups during this period in local scholarship, some
scholars, including G. Joseph of the U. Manchester have suggested that the
writings of the Kerala School may have also been transmitted to Europe around
this time, which was still about a century before Newton.
Charles Whish (1834). "On the Hindu Quadrature of the
circle and the infinite series of the proportion of the circumference to the
diameter exhibited in the four Sastras, the Tantra Sahgraham, Yucti Bhasha,
Carana Paddhati and Sadratnamala". Transactions of the Royal Asiatic
Society of Great Britain and Ireland which was collected and translated into
English 200 Years ago, for Royal Asiatic Society of Great Britain and Ireland.
He wrote about the technicalities of Calculus, Trigonometry and Geometry
related to Astronomy by the said school. This is the first recorded evidence of
our indigenous Calculus, which crossed to overseas.
The Tantrasangraham consists of eight chapters viz.
1.
Madhya Prakaranam (Mean Longitude of planets)
2.
Ravigrahana Prakaranam (calculation of Solar Eclipses)
3.
Sputa Prakaranam (True longitudes of Planets)
4.
Vyateepaatha Prakaranam (Relativity of Sun and Moon)
5.
Chaaya Prakaranam (Gnomonic Shadow)
6.
Dushkarma Prakaranam (Possible errors and corrections)
7.
Chandragraha Prakaranam (Calculation of Lunar eclipses)
8.
Shrungonnathi Prakaranam (earth and Moon Positions)
Tantra Sangraham written in 1501 AD deals with the technics of
finding out solutions for various Astronomical Problems. The original is in
palm leaves in Sanskrit with Malayalam script. These chapters are totally
dependent on Mathematical calculations related to the said branches of
Mathematics. He defines π, in the book as
follows:
व्यासे
वारिधि निहते रूपाहृते व्यास सागाराभिहते l
त्रिशारादी
विशा संङ्ज्ञाभक्त मृणं स्वं पृथक क्रमात कुर्याथ ll
π = 4(1-1\3 +1\5 – 1\7+ 1\9…….) is the exact meaning of the above
sloka. Expressing a finite quantity like π, in an infinite series to gain precession
is the foundational thought of Calculus.
This is in fact to be named as
Madhava Series but unfortunately all the contributions by ‘Bharata’ were
plagiarized by the Westerners and they gave their names to all of them without
feeling any shy for their deeds. Let me bring to your notice a bare fact that
all the contributions by Westerners are given to the world around 15th
Century when the Britishers started occupying ‘Bharata’ and renaissance started
in their country. Why can’t those people start their research work
earlier? Govinda Swami (800-850 AD)
Newton-Gauss interpolation 1750 AD. I can give good number of examples like
this but reading these mathematics formulae will not be palatable for the
ordinary reader who is not conversant with advanced mathematics.
With regard to the quadrature of the circle, initially Whish had
been convinced by his older colleagues, Warren and Hyne, to change his view and
believe that the series had been given to the Hindus by Europeans, but he
reverted to his earlier opinion when he discovered proofs of the results in the
Yuktibhasa. He published a now famous paper On the Hindú Quadrature of the Circle, and the Infinite Series of the
Proportion of the Circumference to the Diameter Exhibited in the Four Sástras, the Tantra Sangraham, Yucti Bháshá, Carana Padhati, and Sadratnamála which was published in the Transactions of the Royal Asiatic
Society of Great Britain and Ireland in 1834. Whish writes in his paper about
the results in the Tantra Sangraham, the Carana Padhati, and the Sadratnamála:-
The approximations to the true value of the circumference with a
given diameter, exhibited in these three works, are so wonderfully correct,
that European mathematicians, who seek for such proportion in the doctrine of
fluxions, or in the more tedious continual bisection of an arc, will wonder by
what means the Hindu has been able to extend the proportion to so great a
length. Some quotations which I shall make from these three books, will show
that a system of fluxions peculiar to their authors alone among Hindus, has
been followed by them in establishing their quadratures of the circle; and a
few more verses, which I shall hereafter treat of and explain, will prove, that
by the same mode also, the sines, cosines, etc. are found with the greatest
accuracy. ... Having thus submitted to the inspection of the curious eight
different infinite series, extracted from Bráhmanical works for
the quadrature of the circle, it will be proper to explain by what steps the
Hindu mathematician has been led to these forms, which have only been made
known to Europeans, through the method of fluxions (Fluxions are the technical
component of Calculus as a subject), the invention of the illustrious Newton.
... it is a fact which I have ascertained beyond a doubt, that the invention of
infinite series of these forms has originated in Malabar, and is not, even to
this day, known to the eastward of the range of Gháts which divides that country, called in the earliest times Céralam, from the countries of Madura, Coimbatore, Mysore, and
those in succession, to that northward of these provinces.
It would be fair to say that Cadambathur Tiruvenkatacharlu
Rajagopal (1903-1978) was the first person to continue Whish's work. Rajagopal
writes in 1949 in [10]:-
A little over a century has elapsed since the first attempt was
made to mark on the map of modern scholarship this virgin continent [Hindu
mathematics]. The person who sighted the unknown coast was, by an odd trick of
time, an English civilian of the Hon East India Company, Charles M Whish by
name. Whish's paper carrying the abbreviated title "On the Hindu
Quadrature of the Circle", submitted to the 'Royal Asiatic Society of
Great Britain and Ireland' on 15th December 1832, did not advertise his
importance as the discoverer of a strange hinterland. There was little in the
title of the paper to assure its readers that the material offered to them had
with difficulty drawn from that stock of mixed mathematics which the children
of Kerala had till then looked upon as its exclusive property; there was
nothing in it which suggested that the author had overcome the exclusivism of
the Keraliyas with the help of their pundits and princes - a help by no means
easy to secure then, for, as we know today, the companions of our author in the
civil service of the Hon East India Company were "fortune-hunting
adventurers lost to all sense of public morality" who did much to alienate
the sympathies of the natives.
Key contributions of Kerala’s School of Astronomy:
Madhava (1350 – 1420) Newton Power Series for Sine and Cosine
1643 – 1727 AD
-Do- Approximations for value Pi -
Newton -do-
-Do- Taylor Series
for Sine and Cosine Functions 1685 – 1731 AD
-Do- Lebiniz series for inverse
Tangents 1646 - 1716
Neela Kantha 1444 – 1544 Lebiniz power series for Pi - Do -
Parameshwara 1360 - 1455
L’Huliar’s Formula of Cyclic Quadrilaterals 1750 - 1840
Govindaswami 800-850 AD Newton-gauss interpolation Newton-gauss
1751 AD
These facts say that Kerala’s contributions to mathematics are
far older than the European discoveries if seen unbiased. It is certain that
these Sanskrit works when travelled across the sea were translated into their
respective languages and may be were owned by their renowned mathematicians.
The Kerala School of astronomy and mathematics was founded by
Madhava of Sangamagrama in Kerala, South India and included among its members:
Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur
Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and
16th centuries and the original discoveries of the school seems to have ended
with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical
problems, the Kerala school astronomers independently created a number of
important mathematics concepts. The most important results, series expansion
for trigonometric functions, were given in Sanskrit verse in a book by
Neelakanta called Tantrasangraha and a commentary on this work called
Tantrasangraha-vakhya of unknown authorship. The theorems were stated without
proof, but proofs for the series for sine, cosine, and inverse tangent were
provided a century later in the work Yuktibhāṣā (c.1500–c.1610),
written in Malayalam, by Jyesthadeva.
The work of the Kerala mathematicians anticipated the calculus
as it developed
in Europe later, and in particular it involves manipulations
with indefinitely small quantities (in the determination of circumference of
the circle etc.) reminiscent
of the infinitesimals in calculus; it has also been argued by
some authors that the
work is indeed calculus already. The overall context raises a
question of possible
transmission of ideas from Kerala to Europe, through some
intermediaries. No definitive evidence has emerged in this respect, but there
have been discussions on the issue based on circumstantial evidence.
I conclude this article with a request to the readers that I
tried to make this attempt above my stature, with inherent zeal than the caliber,
to make them know about the great people of our Motherland ‘Bharata’ and not ‘India’.
References taken from
various texts of eminent authors of our motherland and abroad for authenticity
of the article.
Swasti.
