Lecture on “History of Ancient Indian Mathematics” at the Department of Mathematics

Date: 28^th August 2017

Speaker: Prof. Manjusha Majumdar (Tarafdar), Department of Pure Mathematics, University of Calcutta

Little is known about the achievement of ancient Indian Mathematics. Very few have been printed so far. The contribution can be divided into the following categories:

1. Zero and the place-value notation for numbers
2. Vedic Mathematics and arithmetical operations
3. Sulya-sutras
4. Astronomy
5. Algebra
6. Trigonometry
7. Analysis

1:                    Indian mathematicians have taken “0” as the base a long time ago. Greeks had not Terminology above  .Romans had no terminology above  . But in around  1^st Century B.C, in a Buddhist work “Lalitavistara”, we have  terminology  .Even in “ Yajurveda”, in its description of rituals, in “ Mahabharata” and in “ Ramayana” in the description of statistics and measurements, we have the terminology of the base  and more.

 The number “zero” is the subtle gift of antiquity by the Indians to mankind. The concept itself was one of the most significant inventions. In Sanskrit “sunya” is the word for zero. Europe came to know through Arab, when Md.Musa of Bagdad explained it around 820 AD and for this reason, it is called Indo-Arabic Numerals,whereas , the exact time and name of the inventor is not known . European book (in French) first used “zero” in 1275.

 2:                Patiganita” is the word for Arithmetic in Sanskrit. Actually “Pati” means Board in English and “ganita” means the science of calculation. In Buddhist Literature, it has been mentioned that there are three classes of Ganita, namely “mudra” (finger arithmetic), “ganana” (mental mathematics) and “Samkhyana” (higher arithmetic). Indians\Hindus made remarkable contributions in these fields by saying that there are eight fundamental operations. They are: addition, subtraction, multiplication,division, square ,square-root , cube and cube –root. The names of the great ancient mathematicians whose contributions are remarkable are mentioned below:

Aryabhata I (b 475AD), Bhaskara I (b 528AD), Brahmaupta (b 598AD), Aryabhata II (b 950AD), Bhaskara II(b 1114AD)

3:                    From the times of Vedas , the ritual literature, which gave directions for constructing sacrificant fires, dealt with the measurement and constructions of different kinds of “Vedi” s( altar), thus giving rise to “ sulya sutra”(Geometry). The three most important books are Bodhyana, Apastamba and Katyayana. They are written between 800-500 B.C of which Bodhyana is the oldest and biggest. In these books, we have the instructions for construction of squares, rectangles, parallelograms and trapeziums. In “Bodhyana”, we have the Pythagorean theorem in a different manner. A remarkable achievement was the discovery of the square root of two which is mentioned as 1.4142156…………….  “Katyayana”remarkably exhibits the geometrical knowledge of the human body.Moreover, some of the geometrical statements have been mentioned without proof .Now a days, we call it axioms.

4:                  The contribution of Indian mathematicians is so great in this field that it mesmerises the whole world.  In Sanskrit, “Jyotisa” is the word for Astronomy. Ancient Hindu mathematicians gave 27 formulae by applying which one can say about the exact time while watching the position of stars. Aryabhata I in his famous book “ Aryabhaterya” (499AD) mentioned that the diurnal motion of the heavens is due to the rotation of the earth about the axis. A famous astronomer and astrologer, Varhira wrote a book “ PanchSiddhartika” in 505AD . He gave the accurate value of precession of equinoxes. We are indebted to him for the correct version of Indian calendar.

5:                Ancient Indian Mathematicians mentioned “bijaganita” for Algebra. They said bijaganita deals with symbols which are “avyakta”(unknown). They started studying it separately from Arithmetic in the beginning of the 7^th Century.  Aryabhata I was the first one to give the method for solving quadratic equations and first degree indeterminate equations. He mentioned the approximation value for ? as 3.1416 and gave the formula for+………..+,++……………+. But the remarkable contribution was done by Brahmagupta whose book “Brahma-Sphuta-Siddhanta” (628 AD) was translated by Arab in 770 AD. BhaskaraII called him “ganitachuramoni”. He termed “Kuttaka” (Pulveriser) as algebra. He was the first one to mention that a triangle connected by  += is a right-angled triangle.  All the results in cyclic quadrilaterals, which are taught in high school, are the contributions of Brahmagupta. He pointed out that    a+0=a, a-0=a, a×0=0, =0. His outstanding contribution was the solving of the indeterminate equation. He was alsothe first one to give a rule for interpolation with data at equal intervals. Bhaskara II wrote a book “Siddhanta-Siromoni” in 1150 AD. The arithmetic part of this book is the famous book “Leelavati” which was translated into Persian   by Fyzl,brother of Abul Fazl, under the  command of great Muahal Emperor Akbar in 1587, Algebra portion of this book also was translated  in Persian  by the command of Akbar  in 1634. The other two portions of this book relate to Astronomy where he gave the formula of Sin(A± B) and   and  many more rules for mensuration.

6:             AryabhataI in his book, written in 499 AD, was the first  one to give the table of Sin, concept of Sin. He first introduced rsin, r cos?, where r is the radius of the circle and ? is the angle at the center.

7:                Kerala mathematician Madhaba ( circa 1340-1425 AD)  was the first one  to  develop  infinite series approximation for a range of trigonometric functions , which has today come to be known as Mathematical Analysis, well ahead of Newton, who latter developed this   300 years after Madhaban.

 Let us hope that the new millennium shall spread mathematical knowledge conceived by Indian-born mathematicians.