Subject: [HM] Indian derivation of sine series
From: David M. Bressoud (bressoud@macalester.edu)
Date: Mon Aug 07 2000 - 17:11:14 EDT
Regarding the Indian derivation of the sine and cosine series, I am
confused by what I find in both the scholarly and popular literature.
Below, I summarize what I have found. I ask for help: What is currently
considered reliable, what is questionable, what are the current best
guesses for dates and attributions?
Rajagopal & Venkataram ("The sine and cosine power-series in Hindu
Mathematics", J. Royal Asiatic Society of Bengal. Science. 1949) assert
that these series are in Nilakantha's Tantrasamgraha of ca 1500 and even
quote it. The explanation of the derivation is given in the Yukti-Bhasa
which they date to 1750, but in an addendum by K. M. George, stylistic
arguments are used to argue that the Yukti-Bhasa must have been created in
the early 16th century. No authorship is proposed for the Yukti-Bhasa at
this point.
Rajagopal & Rangachari ("On an untapped source of medieval Keralese
mathematics", Arch. for Hist of Exact Sci. 1978) now assert that these
series are not in the existing manuscripts of the Tantrasamgraha, but that
they are to be found in a commentary, the Tantrasamgraha-vyakhya, and argue
that they probably were present in the full text of the Tantrasamgraha,
which no longer exists. They assert that, in spite of its name, the
Tantrasamgraha-vyakhya is primarliy a commentary on the Yukti-bhasa (and
thus, presumably, of later origin), that the author of the T-vyakhya was
Sankara Variyar, and that this implies that the T-vyakhya was written prior
to 1534, and hence one would expect that the Yukti-bhasa was also written
before 1534. They also cite a 1958 paper by K.V. Sarma that identifies
Jyesthadeva as the author of the Yukti-bhasa, and they give his dates as c.
1500 - c. 1610 (sic). They also state that the series for the sine appears
in a late work of Nilakantha, the Aryabhatiya-bhasya, where, as in several
other places, it is attributed to Madhava, ca 1400. Since Nilkantha died
around (is it known exactly?) 1545, this appearance of the series would
also appear to be reliably first half of 16th century.
Now to the popular literature.
Bose, Sen, and Subbarayappa (A Concise History of Science in India. Vol on
Source Materials, Astronommy, and Mathematics. Indian National Academy,
1971) state (p 201) that these series can be derived from rules given in
the Karanapaddhati believed to have been written by Putumana Somayajin in
the 15th century (this is the only reference I have found to this, where
does it come from and is there anything further?). They state that they can
be found in the Tantrasamgraha of 1500 as well as in the Yuktibhasa which
they date as 1639 (sic).
Roy ("The discovery of the series formula for pi by Leibniz, Gregory and
Nilakantha", Math Mag, 1990) asserts that these series are found in the
Tantrasamgraha, the Yukti-bhasa, and the T.-vyakhya.
Joseph (The Crest of the Peacock, 1994) does not address the question of
whether or not the sine and cosine series are to be found in the
Tantrasamgraha. He gives the date of the Yukti-bhasa as ca. 1550 and the
dates of Sankara Variyar as ca 1500-1560
Katz (A History of Mathematics, 1993) says that these series appear in the
Tantrasamgraha-vyakhya which he dates to c. 1530 and the Yukti-bhasa
written by Jyesthadeva whom he dates as (1500-1610) (significantly dropping
the c.). He mentions the Tantrasamgraha, but is careful not to assert that
the series can be found there.
I am not surprised that there is so much conflicting information, but what
is known and what is currently considered most reliable?
David
David M Bressoud
DeWitt Wallace Professor and Chair bressoud@macalester.edu
Mathematics and Computer Science Department 651-696-6559
Macalester College, 1600 Grand Avenue FAX: 651-696-6518
Saint Paul, MN 55105, USA http://www.macalester.edu/~bressoud
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