MODULE 3 - BHASKARACHARYA (1114 AD – 1185 AD)

BHASKARACHARYA (1114 AD – 1185 AD)

BHASKARA – II

Born	1114 AD
Died	1185 AD
Residence	Vijayapura. Ujjain
Nationality	Indian
Fields	Mathematics, Astronomy
Institutions	Astronomical observatory
Philosophers	Varahamihira,Bramagupta

          Bhaskaracharya otherwise known as Bhaskara II was one of the most powerful and creative mathematicians of ancient India. He was also known as Bhaskara the Learned. He was born in 1114 AD in Vijayapura. His father, Mahesvara, himself was a famous astrologer. In many ways Bhaskaracharya represents the peak of mathematical knowledge of 12^th century. He was the head of the astronomical observatory at Ujjain.

BHASKARA –  BOOKS

There are six well known works of Bhaskaracharya. They are

•          Lilavathi - Mathematics

•          Bijaganita - Algebra

•          Siddhantasiromani – first part mathematical astronomy and second part sphere

•          Vasanabhasya

•          Karanakutuhala

•          Vivarana

However, among the six works of Bhaskaracharya, the first three are more interesting from the point of view of mathematics.

Lilavathi contains 13 chapters and covers topics such as Définition, Mathematical terms, Interest, Arithmetical progression and Geometrical progression , Plane geometry , Solid geometry , Shadow of the gnomon , Kuttaka and combinations.

Bijaganita contains 12 chapters and covers topics like as Positive and negative numbers, Zero , Surds, the kuttaka , Indeterminate quadratic equation with more than one unknown , Quadratic equation with more than one unknown , Operations with products of several unknown.

Siddhantasironmani is a mathematical astronomy book compiled in two parts. First part contains twelve chapters dealing with topics such as Longitudes of the planets, True longitudes of the planets, 3 problems of decimal rotation , syzygies, Lunar eclipse , Solar eclipse, Latitudes of the planets , Rising and setting , Moon’s crescent , Conjunction of the planets  , The pates of sun and moon

Second part of siddhantasiromani contains 13 chapters on the sphere. The topics such as: Praise of study of the sphere , Nature of the sphere , Cosmography and geography , Planetary mean motion , Eccentric epicyclic model ,The armillary sphere , Spherical trigonometry , Ellipse calculations , First visibilities of planets , Calculating the lunar crescents , Astronomical instruments , Problems of astronomical calculations

 

Contributions:

Negative Numbers:

™  Bhaskaracharya was known for his treatment of negative numbers with he considered as debts or losses, and also for his treatise on arithmetic and measurement.

™  Bhaskaracharya also handled efficiently arithmetic involving negative numbers.

™  In Bijaganita placing a dot above them denotes negative numbers.

Infinity & Zero:

™  He for the first time brought the idea of infinity while dividing a number by zero.

Zero rules:

™  He was sound in addition, subtraction and multiplication involving zero but realized that there were problems with Brahmagupta’s idea of dividing by zero.

Ü    A + 0 = A

Ü    A – 0 = A

Ü    A x 0  = 0

™  He understood about zero and negative numbers and he knew that x² = 9 had two solutions.

Progression:

™  He was aware of arithmetical and geometrical progression and explains examples.

Sphere:

™  He found formula for finding the area and volume of sphere given below:

Area of sphere = 4 x area of a circle.

Volume of a sphere = area of a sphere x 1/6 of its diameter.

Trigonometry:

™   He seems more interested in trigonometry. Among the many interesting results given by bhaskaracharya are:        

sin (a + b) = sin a cos b + cos a sin b

        And sin (a - b) = sin a cos b - cos a sin b.

Lilavati:

™   Bhaskaracharya gave two methods of multiplication in Lilavati.

™   It is argued that zero used by bhaskaracharya, in his rule (a.0)/0 = a given in Lilavathi, is equivalent to the modern concept of a non-zero “infinitesimal”.

Other works:

™   He has used the kuttaka method of solving indeterminate equations.

™   He had explained the concepts of permutation combination with examples.

™   In differential calculus he was the first mathematician who presented examples related to differential coefficient.

™   He originated the fundamentals of Rolle’s theorem.

™   He knew about inverse proportions and rule of the three.

          Bhaskaracharya’s innumerable contribution had earned him an outstanding position among the ancient Hindu mathematicians. He dies in Ujjain in 1185.