The Classical
Era Continues
For centuries after Bhaskara, the classical
era continued. The Ganita Kaumudi of
Narayana Pandita is a fascinating book, with an interesting structure and
several novelties. Like Mahavira, he gave both rules and a series of examples. He
expanded Arybhata’s formula for sum of series of numbers, their squares, and
cubes, to any arbitrary power.
Effectively, he provided the formula
(Sn)r = [n(n+1)(n+2)…(n+r)] / [1.2….(r+1)]
Narayana also expanded on the meru prastaras
(Pascal’s triangles) of Pingala and Hemachandra into more elaborate versions.
Permuatations, combinations, areas and volumes of more compelex geometric
shapes were other areas of exploration and explanation.
His famous novelty was magic squares, squares
filled with numbers so that each row and column adds up to the same number. He
explained procedures to create magic squares of desired sizes, how to fill them
up for a given total, and certain fascinating variations, including use of
negative numbers. He extended this to magic rectangles, and other figures, like
squares within squares, triangles within triangles etc. He also devised many
more such shapes like vajra
(diamond), padma (lotus), padmavrtta(lotus circle), manDapa(canopy), sadashra (hexagon), dvaadashaashra
(dodacehedron) etc filled with such numbers. These are explorations with
perhaps no practical applications, especially in astronomy.
Later Jaina mathematicians Dharamanandana and
Sundarasuri, continued explorations on magic squares and similar arrangements.
But from the fourteenth century, Kerala became
the locus of several new siddhaantas, bhaashyas, karanas etc., that it is now
called the Kerala school of mathematics.
The Kerala
school
Shortly after Aryabhata, a mathematician
called Haridatta had composed a work title ParahitaGanita
based on Aryabhatiyam. After that we
seem to have a vaccum in Kerala until Madhava (c 1340 AD) of Sangamagrama
(Kudalur in Malayalam). What followed is an astonishing continuity of the guru-shishya
parampara from Govinda Bhattatiri to Rajaraja Varma.
Madhava was famous as a skilled instrument
maker. Indian historians of mathematics consider him the pitamaha of the Kerala
school. Perhaps his most famous contribution is sums of infinite series. To calculate
the circumference (paridhi) of a
circle, said Madhava, we must multiply the diameter (vyaasa) by four times one minus tri-sharaa-aadi-vishama-samkhyaa-bhaktam-rNam-svam-prtak.
A phrase of compactness, which Aryabhata would have enjoyed. In other words, a sequence of odd
(vishama samkhyaa) denominators (bhaktam) starting (aadi) with three (tri) and
five (sharaa), negative(rNam) svam(positive) alternately(prtak). The word sharaa here is a bhutasamkhya (see third essay) word for the five arrows of
Manmatha, whose archery takes precedence over Rama and Arjuna and even
Tripurantaka, in this case.
paridhi = 4 * vyaasa * (1 -1/3 + 1/5 - 1/7 + 1/9 - 1/11....)
The sum of the series in the brackets adds up
to π/4, and is famous as the Gregory Leibniz series.
Madhava’s series was quoted and a proof (upapatti) also given a century later by
Jyeshtadeva.
Parameshvara – Like
Brahmagupta providing a sphuTam to Paitamaha Siddhaanta, Parameshvara
observed that over time predictions of earlier astronomers did not agree with
observed positions based on calculations. In such a situation, he observed, one
must adjust one’s methods and caluclations, because planets and stars will
confirm to He titled his book Drg Ganitam.
This title, which means Observed Calculations, is a popular phrase for jyotisham in south
India, though the author himself has faded from public memory. His shishya
NilakanTha referred to him as Paschimaam
Bodhi, the western scholar.
Nilakantha Somayaaji
– More unknown than even Brahmagupta, NilakanTha was a polymath like Varahamihira.
He was a scholar of Shad darshana
(the six philosophies of Hinduism, and also in vyaakarana, chandas, the Bhagavata and various such litereature. He also studied Vedanga Jyotisha, Pancha Siddhaantika,
Brhat Samhita etc. This historical curiosity and scholarship may have shined in
other scholars, too, but in Nilakantha we have contemporary evidence. He was
also a prolific composer, of several texts.
He was a friend a Sundararaja, a jyotisha of
the neighboring Tamilnadu, and took the effort to compile a written list of
answers for questions posed by the former, compiled into a book called Sundararaja Prashnottara. Aficionados of
European science may be reminded of the extensive correspondonce of Franklin,
Newton, Darwin, Humboldt etc.
A ninth century mathematician called Virasena
in his commentary Dhavala gave this
equation, that the sum of all powers of 1/4 is 1/3. One sees the reflection of
this in Madhava’s several infinte series. Nilakantha questioned this apparent
absurdity. How does the sum of this infinite series increase to that finite
value(1/3), and that it reaches finite value?
He reasoned and explained it by deriving the
following sequence of results
As we add more terms, argued Nilakantha, the
difference between 1/3 and the powers of 1/4 become extremely small, but never
zero, unless we add terms upto infinity. In the 20th century,
Ramanujan reveled in such series.
Quasi Heliocentric theory
NilakanTha’s questioning of an assumption of planets’ latitudes(vikshepa), and his subsequent discovery was
truly astronomical, pardon the pun. From the siddhantas through Bhaskaracharya,
all astronomers used a slightly different method to calculate the latitudes of
Mercury (Budha) and Venus (Shukra), than they did for the other planets. This
niggled most of them, as “inappropriate”, especially Bhaskara, who then
consoled himself with Prthudaka Svami’s exlpanation. But NilakanTha questioned
this acceptance, and modified his computation, and effectively the orbital
model for these planets. He came to the conclusion that these two planets
revolve around the Sun (but in his model, the Sun still revolves around the
Earth). The geometrical argument is too complicated not only for this article,
but even perhaps for those who are not astronomers, so I will only present a
visual illustration of his model here.
Effectively, NilakanTha formulated a
quasi-heliocentric theory : sun centric for inner planets, and earth centric
for everything else.
Jyeshta Deva a junior
of Nilakantha wrote a work called Yuktibhaasha,
in Malayalam, not Sanskrit. He extensively dealt with several proofs and also
developed the theory of infinetesimals to greater refinement and
sophistication.
Modern Era
The modern era was introduced gradually into
India, with the advent of the East India company, and the influx of European
scientific discoveries it brought along. I’ll refrain from the delving into the
astronomical the Europeans themselves conducted here, including the
establishment of the Madras observatory, the Madras meridian, observations of
Venus transit, the discovery of Helium etc.
Sawai Jaisingh’s several Jantar Mantar are
perhaps the most famous observatories in India, and really belong to the modern
era. He was also instrumental in commissioning translations of Euclid and
Ptolemy from Persian.
The most fascinating ganaka or siddhanta
acharya of the modern era was Pathani Samanta Chandrasekhar, of Orissa. Utterly
unaware of the European advances in astronomy including telescopes,
electricity, gravity and optics, or even of the Madras observatory, and
apparently not even aware of the Kerala school, he independently
discovered the quasi helio centric theory of NilakanTha Somayaji. He also made
several corrections and improvements to Bhaskara’s Siddhanata, using only the
naked eye, besides develiping his own new instruments. His book Siddhanta
Darpana, with 2500 slokas was published by PC Ray . He is only jyotishaacharya
of whom we have a genuine portrait picture. The Bhubaneshvar obseratory is
named after him.
Orientalists
Sir William Jones, judge of the Supreme court
of Bengal, founder of the Asiatic Society of Bengal, discoverer of the
Indo-European family of languages, translator of Manu Smriti and Shakuntalam
into Sanskrit, rediscoverer of Pataliputra, ad infinitum, also declared that
India gave the world algebra. Why we learn in school about FaHien and Hiuen
Tsang, al Beruni and ibn Batuta, but nothing of this giant of Indian
scholarship, I’ll never know. Some of us Indians possess the same myopic
prejudice against such European stalwarts as the haughty colonialists did of us
lesser Indians. It may also be cognitive dissonance.
The Asiatic Society and such similar
institutions like the Madras Literary Society produced extraordinary
scholarship about several aspects of Indian history and culture, including
astronomy and mathematics. Henry Thomas Colebrooke, George Thibaut, John
Playfair, Burnell, Burgess, and such other European scholars translated,
analyzed and compiled the history of Indian sciences and exposed them to the
world., Charles Whish, and his collaborators John Warren and George Hyne played
a significant role in bringing to light the Kerala school in the early
nineteenth century. This was also the period when English effectively replaced
Sanskrit as the primary language of science in India. The spectacular advances
brought about by the Industrial revolution, and subsequent scientific advances
in Europe, and the corresponding rise of European intellectual and military
supremacy, drowned out the accomplishments of Indians in the past (and other
cultures as well). The rest of the world is catching up in application of technology,
but is stall far behind in the curiosity and spirit that drove discoveries and inventions.
That is now changing.
European orientalists did not work in
isolation. Indian pandits, like Sudhakara Dwivedi, Chidambaram Iyer, and
numerous others helped translate Sanskrit texts into English, French and other
languages.
 |
| Sir William Jones |
In the twentieth century scholars like
Bibhutibhushan Dutta and Avdesh Narayan Singh, TS Kuppanna Sastri, KV Sarma, KS
Shukla, Rangacharya and others have translated books, and given us various
detailed studies of such works also. Unfortuntately, these remain confined to
the small community of scientists who know Sanskrit. |
| KV Sarma |
The future
In my first essay in these series, I presented
a quiz, contrasting our ignorance of Indian scientists and their contributions
versus our knowledge of European scientists. I hope some of these essays have
addressed that lacuna: we know a little about Brahmagupta and Nilakantha. There
was much science in the investigations of our star gazers, than taught of in
our philosophy. I hope that it has provoked curiosity about Indian history,
especially of science and technology, and a desire to at least seek some of
this information, of which we have zero exposure in the public education
system. I hope also that we as a society, try to learn about sciences of
various ancient cultures, such as China, Egypt, Olmecs-Mayans, Sumeria, etc. It
is fascinating that Mayans independently discovered the zero, but not the place
value system, for example. And our 24 hour clock is based on Egyptian
astronomy.
I am not a mathematician, an astronomer, or a
historian; I don’t know Sanskrit, except for a smattering of vocabulary,
especially technical and scientific. If even I can learn enough to write a
series like this, what can you not do?
-----------------
This essay was first published in Swarajya magazine
For the entire series click this link --> Indian Astronomy and Mathematics
References
1Facets of Indian astronomy, KV Sarma
2 Studies in Indian Mathematics and
Astronomy (Selected articles of Kripa Shankar Shukla) – Kolachana, Mahesh,
Ramasubramanian
500 Years of TantraSangraha, MS
Sriram, MD Srinivas, K Ramasubramaniam, 2002
Lecture on Samanta Chandrasekhar, by
Dr Prahlad Chandra Naik , Seminar in Univ of Madras, February 2011.
The Discovery of Madhava series by
Whish, Ganita Bharati Vol 32, 2002
NPTEL Lectures on Indian Mathematics by Profs MD Srinivas, MS Sriram, K Ramasubramanian
