Newsgroups: soc.culture.tamil
Date: Wed, 12 Aug 1992 22:58:04 -0400 
From: "Arun K. Gupta" <ag3l+@andrew.cmu.edu>
Subject: Ancient Indian mathematics


The following was the result of some conversation on sci.math that
I had. It mentions a math. book in Malayalam. Presumably, the math.
tradition mentioned below wasn't isolated. It would be interesting
to see if it left any traces elsewhere in the south. Gheverghese
Joseph speculates that Ramanujam is somehow connected to this
tradition.

akg

*****************************************************
"Like the crest of a peacock, like the gem on the head of a snake,
so is mathematics at the head of all knowledge."

-Vedanga Jyotisa (c. 500 BC)
*******************************************************

akg:

In the library, I stumbled across a book: "The Crest of the Peacock:
Non-European Roots of Mathematics", by George Gheverghese Joseph, of
the University of Manchester.  (I.B. Tauris & Co. Ltd, 1991).

The book makes some extraordinary claims: here is one:

Madhava (also known as `Golavid' or Master of Spherics) of Sangamagramma
(a village near Cochin in Kerala) (c1340-1425) was "probably the greatest
of the Indian medieval astronomer- mathematicians".

Only his astronomical treatises survive, for his mathematics the reports
by contemporaries and successors must be relied on.

In 1835, Charles Whish published an article referring to four works of
Madhava's contemporaries or followers:

 C.M. Whish (1835) `On the Hindu quadrature and infinite series of the
proportion of the circumference to the diameter exhibited in the four
Sastras, the Tantrasangraham, Yucti-Bhasa, Carana Padhati and Sadrat-
namala', Transanctions of the Royal Asiatic Society of Great Britain
and Ireland, 3, part 3, 509-23).

Even so, "none of the results has as yet percolated into the standard
Western histories of mathematics".

Some of the results attributed to Madhava are: (translated to modern
notation)

(i) Gregory's (1667) series for the inverse tangent:
                 3     5
                x     x
arctan(x)= x - ___ + ___ - ..., x <=1
                3     5

(ii) the Euler series (x = 1 in the above)

pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...

and  ( x = 1/sqrt(3))

pi = sqrt(12) ( 1 - 1/(3 * 3) + 1/(5*3^2) - 1/(7*3^3) + ....)

(iii)the approximation

pi/4 = 1 -1/3 + 1/5 - ... (-1)^(n-1) / ( 2 n -1) + (-1)^n (n / ( 4n^2 +1));

The further improvement is suggested, replacing the last term by

  (n^2 + 1) / ( n ((n+1)^4 + 1))

Madhava (probably with this formula) calculated a value of pi of
3.141 592 653 59 (he calculated the circumference of a circle of
diameter 900 000 000 000 to be 2 827 433 388 233 ).

(iv) Madhava also gave the power series for the sine

sin(x) = x - x^3 / 3! + x^5/ 5!  - ...

(attributed to Newton, 1676) and the cosine.

**********
Comments on this would be most welcome, especially from historians
of mathematics.
-arun gupta
ag3L@andrew.cmu.edu
*********************************************
From: tee@mat.aukuni.ac.nz
Subject: Madhava

Dear Dr Gupta,
              The reports of Madhava's achievements in trigonometric series
made by George Gheverghese Joseph are accurate.

        After Charles Whish published his paper in 1835, very little
attention was paid to it for 115 years. One principal reason is the very
great difficulty of understanding his Sanskrit source texts. Traditional
Indian scientific texts were written as concise statements in Sanskrit
verses (even trigonometric tables!), and proofs were very rarely given in
mathematical texts.

        Another major reason was that, in the hey-day of European
Imperialism, very few Europeans had any interest in intellectual
achievements by anyone other than Europeans. Between 1835 and 1950, very
few of the Europeans who did make any serious study of Indian culture knew
enough science to be able to understand Indian scientific texts.

        B. Datta & A. N. Singh wrote the valuable "History of Indian
mathematics". Volumes 1 & 2 (arithmetic & algebra) were published in the
1930s; but their volume 3 (on geometry and trigonometry) was not published
until the 1970s. 

        The texts reported by Charles Whish included the Yucti-Bhasa,
written in Malayalam (rather than Sanskrit), apparently in the 16th
century. (In traditional Indian culture, dates are frequently very
difficult to determine.) In 1952 the text of Yucti-Bhasa was published,
with English translation. The Yucti-Bhasa is a commentary on several
earlier texts by Madhava, Nilakantha Somajin et alia. Those cryptic
Sanskrit verses had mostly been incomprehensible, until the Yucti-Bhasa was
found to give detailed proofs of several power series for trigonometric and
inverse trigonometric functions.

        One of the best accounts of Indian mathematics (including those
power series) is given by A. P. Yushkevich in "Istoriya matematiki v
sredniye veka" (History of mathematics in the middle ages), FizMatGiz,
Moscow, 1961. (A German translation has been published, as: A. P.
Juschkewitsch, "Mathematik in der Mittelalter".) A brief account is given
in: D. M. Bose, S. N. Sen & B. V. Subbarayappa (editors), "A concise
history of science in India".

        A technical study of Indian geometry & trigonometry, with Sanskrit
texts included, is given by Amma T. Sarasvati, "Geometry in ancient and
medieval India".
                                        Yours sincerely,
                                                        Garry J. Tee,
                                     University of Auckland, New Zealand.


Reply-To: mff@ukc.ac.uk (M.F.Fuller)

I think it must have been the author of "The Crest of the Peacock" whom I
heard a week or so ago on BBC Radio 4's "Start the Week" with Melvyn Bragg -
not a programme I usually associate with mathematics. The angle of particular
focus there was the cultural chauvinism of the Anglo Saxons in overlooking
the mathematical achievements of what we now label as the Third World.

For another interesting insight into Indian mathematics, may I recommend Ian
Hacking's "The Emergence of Probability", CUP, 1975. Chapter 1 of this points
out that in the Mahabarata, which probably reached its present form by AD 400,
the story of Nala suggests that - a millenium before this was dimly seen in
Western Europe - Indians were aware of the link between the analysis of dice
games and sampling. When Rtuparna estimates the number of leaves and fruit on
a tree from inspection of a single twig, and Nala counts all night to check
the accuracy of the estimate, Rtuparna explains his successful prediction with
the words "I of dice possess the science, and in numbers thus am skilled."

Hacking also mentions a ninth century AD example of a 'Dutch book' - the
careful
placing of bets at odds guaranteed to provide a profit, no matter what the
outcome - in the writings of Mahaviracarya.

Mike Fuller
- statistician marooned in Canterbury Business School, University of Kent,
Canterbury, Kent, CT2 7PD, England (email: mff@ukc.ac.uk - in UK mff@uk.ac.ukc)

P.S. I did not manage to see Peter Brook's production of the Mahabarata, 
performed in Glasgow and elsewhere a few years ago. If anyone did see it, and 
is able to recall whether the Nala story was included or excluded, I would be
interested to know.

From: gedgar@magnus.acs.ohio-state.edu (Gerald A Edgar)

> Some of the results attributed to Madhava are: (translated to modern
> notation)
>
> (i) Gregory's (1667) series for the inverse tangent:
>                 3     5
>                x     x
>arctan(x)= x - ___ + ___ - ..., x <=1
>                3     5

See also the award-winning article by Ranjan Roy,
"The Discoverty of the Series Formula for Pi by Leibnitz,
Gregory, and Nilakantha", Mathematics Magazine 63 (1990) 291--306.

Nilakantha (c.1450--1550) wrote several such series.  He attributes
the series for the sine to Madhava, but gives no attribution for
the arctan series, above.

Roy traces several independent discoveries of the series in his interesting
article.  He concentrates on Europe and India, but also mentions a
Japanese series for pi dating from 1722.  Roy's closing sentence:

"However, the independent discovery of the infinite series [for pi]
 by different persons living in different environments and cultures
 gives us insight into the character of mathematics as a universal
 discipline."
-- 
  Gerald A. Edgar                Internet:  edgar@mps.ohio-state.edu
  Department of Mathematics      Bitnet:    EDGAR@OHSTPY
  The Ohio State University      telephone: 614-292-0395 (Office)
  Columbus, OH 43210             -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)


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