MAT 314: Neither Newton nor Leibnitz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala

7 — 11 March 2005

Old Main 403 Canisius College: 5:00 — 8:00pm Daily
Instructor:   Dr Sarada Rajeev
Professor of Physics
University of Rochester
Rochester, New York
rajeev@pas.rochester.edu

Course Archives
All files below are Adobe Acrobat (.pdf).  File size as indicated.

 Course Abstract for MAT 314:  There was a bitter controversy involving Newton and Leibnitz in the seventeenth century over the invention of calculus. Although Newton remains the undisputed giant of the field and the founder of modern physics, it has been since discovered (largely through the historical scholarship of K. V. Sharma) that many of the basic ideas of calculus were known previously to astronomers of a South Indian school. They knew of the theory of infinite series, notions of convergence, differentiation, and iterative methods for solution of non-linear equations. This school, founded by Madhava of Sangamagrama and included as its prominent members Neelakanta Somayaji, Parameswara, Jyeshtadeva and Achyuta Panikkar, flourished between the 13th and 16th centuries-but has its intellectual roots with Aryabhatta who lived in the 5th century.

We will review their knowledge of (pre-Keplerian, epicyclic) astronomy and the problems that they were attempting to solve that led them to such revolutionary mathematical discoveries. We will get a glimpse into the contents of the first calculus textbook in the world, the Yuktibhasha (in the South Indian language Malayalam) written by Jyeshtadeva in the 16th century. The spice economy of Kerala and its unique social structure (centered around the temple and its Hindu religious rites) in which this community of scholars existed will also be described.

We will then attempt to place the work of the Kerala school in the modern context: the definitive solution of the ancient problems of celestial mechanics by Newton and his successors and the ever expanding applications of calculus to every field of science.

Some References for MAT 314:

  1. A popular exposition assuming little mathematics knowledge: George Gheverghese Joseph, The Crest of the Peacock, Princeton University Press (2000)
  2. A scholarly treatise with references to original manuscripts and translations of key passages (hard to get in bookstores): K. V. Sharma, A History of the Kerala School of Hindu Astronomy Vishveshvaranand Institute, Hoshiapur (1972).
Two standard treatises on the history of related subjects:
  1. V. S. Varadarajan, Algebra in Ancient and Modern Times, Providence, RI: Amer. Math. Soc., 1998.
  2. B. L. van der Waerden Geometry and Algebra in Ancient Civilizations Springer-Verlag (1983)
Scholarly articles by historians of mathematics:
  1. R. C. Gupta, Madhava’s and other medieval Indian values of pi, Math. Education 9 (3) (1975), B45-B48; Madhava’s power series computation of the sine Ganita 27 (1-2) (1976), 19-24.
  2. D. Gold and D. Pingree, A hitherto unknown Sanskrit work concerning Madhava’s derivation of the power series for sine and cosine Historia Sci. No. 42 (1991), 49-65.
  3. Hayashi, T. Kusuba and M. Yano, The correction of the Madhava series for the circumference of a circle, Centaurus 33 (2-3) (1990), 149-174.
  4. C. T. Rajagopal and M. S. Rangachari, On an untapped source of medieval Keralese mathematics, Arch. History Exact Sci. 18 (1978), 89-102; On medieval Keralese mathematics, Arch. History Exact Sci. 35 (1986), 91-99.
  5. Original text on Calculus, reprinted by Kerala University from a manuscript in the Oriental Manuscripts Library: Jyeshtadeva, Yuktibhasha (in Malayalam c. 1550 AD).
  6. Web site on the History of Kerala: http://www.keralahistory.ac.in
Biographical Sketch of Professor Rajeev:
Professor Rajeev received his B.Sc. in Physics from the University of Kerala, Trivandrum, India (1979), and his Ph.D. in Physics from Syracuse University (1984). He was a postdoctoral fellow at the Massachusetts Institute of Technology (1984-87) before joining the University of Rochester as an Assistant Professor of Physics in 1987. He was promoted to Associate Professor in 1993, and to Full Professor in 2000. Rajeev has held visiting appointments at the Institute for Advanced Study, Princeton (1991), the Research Institute for Theoretical Physics, Helsinki (1991), and the Mittag-Leffler Institute, Stockholm (1998).

Professor Rajeev works on various problems in theoretical physics arising from “High Energy Physics” as well as other fields such as string theory, nonlinear optics, hydrodynamics and quantum information theory. The focus of his most recent work has been on non-commutative geometry and probability theory, which has applications to understanding Yang-Mills fields as well as to string theory and even classical problems in hydrodynamics. Solitons in field theories of High Energy Physics are another theme of Rajeev’s research.