Madhava of Kerala discovered infinite power series expansions for trigonometric functions like sine, cosine, and arctangent in the 14th century, over 250 years before similar discoveries were made in Europe. These include the Madhava-Gregory series for pi and the Madhava-Newton power series for sine and cosine. Madhava founded the influential Kerala school of astronomy and mathematics. Several historians and mathematicians are quoted praising the significance and innovation of Madhava's work, though it went undiscovered by Western scholars for many years.

1. 7.MADHAVA-GREGORY SERIES.
Part 1: Quotes from four admirers.
Part 2: Sanskrit verses and meanings.

2. FOUR ADMIRERS
 1.Charles M.Whish.
 2.Kim Plofker.
 3.Frits Staal.
 4.George Joseph.

3. CHARLES M.WHISH
 English civil servant in the Madras Establishment of
East India company.
 Article on “Hindu quadrature of the circle and the
infinite series”.

4. QUOTE
 “Kerala mathematicians had laid the foundation for
a complete system of fluxions and their works
abound with fluxional forms and series to be found
in no work of foreign countries”.

5. QUOTE
 “The mathematical heritage of the Indian
subcontinent has long been recognized as
extraordinarily rich. For well over 2500 years
Sanskrit texts have recorded the mathematical
interests and achievements of Indian scholars,
scientists, priests and merchants. Hundreds and
thousands of manuscripts in India and elsewhere
attest to this tradition. . . . Power series in the
Kerala school have become standard episodes in
the story told by general histories of mathematics”-
Kim Plofker.

6. THE BOOK FROM WHICH FOUR PASSAGES ARE
QUOTED.

7. CHAPTER 7. THE SCHOOL OF MADHAVA IN
KERALA.
 “The crest-jewel of the Kerala school is generally
considered to be the infinite series for trigonometric
quantities discovered by its founder Madhava.” –
Kim Plofker, page 222.

8. ABOUT THE AUTHOR KIM PLOFKER
 Trained by David Pingree.
 Born in 1964.
 American historian of mathematics, specializing in
Indian mathematics.
 She is currently a visiting professor at Union
College in Schenectady.

9. QUOTE
 “Probably the most famous school in Indian
mathematics, and the one that produced many of
its most remarkable discoveries, is the
guruparampara or chain of teachers originating with
Madhava in the late 14th century and continuing at
least into the beginning of the 17th.”- page 217.

10. QUOTE
 “Madhava was a Brahmana of the caste known as
Emprantiri, his home was in Sangamagrama,
modern Irinjalakuda. His most famous
mathematical achievements are the Madhava-
Leibnitz series for pi/4 and the Madhava-Newton
power series for the sine and cosine”. – page 218.

11. QUOTES
“Infinite power series that are expansions of pi and the Trigonometric functions
sine,cosine and so forth were discovered by Madhava in the late 14th century
almost three centuries before they were discovered in Europe by Gregory”
“India’s strength lies in abstract theory as is evident from her Mathematics and
Linguistics” – Frits Staal.

12. FRITS STAAL
Professor Emeritus of Philosophy and of
South and Southeast Asian Studies,
University of California at Berkeley.
He received Ph.D in Indian Philosophy and
Sanskrit from University of Madras and
Banaras Hindu University.

13. SOME BOOKS AUTHORED BY FRITS STAAL
 Ritual and Mantras, Rules without neaning.
 Universals: Studies in Indian logic & linguistics.
 Discovering Vedas: Origins, Mantras, Rituals…
 A Reader on the Sanskrit Grammarians.
 The fidelity of oral tradition and the origins of
science.

14. QUOTE
Indians predated Newton discovery by250 years.
“Dr.George Ghevergese Joseph from the University of
Manchester says the Kerala school identified the infinite
series, one of the basic components of calculus, in about
1350. The discovery is currently and wrongly attributed ti
Sir Isaac Newton and Gottfried Leibniz”. – phys.org

16. FIRST PART ENDS.
Second part starts.

17. MADHAVA-GREGORY SERIES

18. THE SERIES IN MODERN NOTATION
 1 – (1/3) + (1/5) – (1/7) + . . .
 Sin x = x – (x^3)/3! + (x^5)/5! - …
 Cos x = 1 – (x^2)/2! + (x^4)/4! - …
 X = tan x – (1/3)(tan x)^3 + (1/5)(tan x)^5 - …

19. QUOTE
 “One example I can give you relates to the Indian
Madhava’s demonstration, in about 1400 A.D. of
the infinite power series of trigonometric functions
using geometrical and algebraic arguments. …
Madhava’s achievements were ignored by western
historians, presumably at first because they could
not admit that an Indian discovered the calculus,
but later because no one read Whish’s article” –
David Pingree (1992).

20. DAVID PINGREE
 America’s foremost historian of exact sciences in
antiquity.
 1933-2005.

21. EDMUND FREDERICK ROBERTSON
• Edmund Frederick Robertson FRSE B.Sc M.Sc
Ph.D., born on 1 June 1943, St Andrews, Scotland,
is a Professor emeritus of pure mathematics at the
University of St Andrews.
• He is one of the creators of the noted MacTutor
History of Mathematics Archive along with John J.
O'Connor. He has written over one hundred
research articles, mainly in the theory of groups and
semigroups. He is the author or co-author of
seventeen textbooks.
• In 1998 he was elected a fellow of the Royal
Society of Edinburgh.

22. QUOTE
 “Madhava was far more innovative than any other
Indian mathematician producing a totally new
perspective on mathematics”. – Canner &
Robertson.